Durston and Craig on an infinite temporal past . . .

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In recent days, the issue of an infinite temporal past as a step by step causal succession has come up at UD. For, it seems the evolutionary materialist faces the unwelcome choice of a cosmos from a true nothing — non-being or else an actually completed infinite past succession of finite causal steps.

Durston:

>>To  avoid  the  theological  and  philosophical  implications  of  a  beginning  for the  universe,  some  naturalists  such  as  Sean  Carroll  suggest  that  all  we  need  to  do  is  build  a  successful  mathematical  model  of  the  universe  where  time  t runs  from  minus  infinity  to  positive  infinity. Although  there  is  no  problem  in  having  t run  from  minus  infinity  to  plus  infinity with  a  mathematical  model,  the real past  history  of  the  universe  cannot  be  a  completed  infinity  of  seconds  that  elapsed,  one  second  at  a  time. There  are at  least  two  problems.  First,  an  infinite  real  past  requires  a  completed  infinity, which  is  a  single  object and  does  not  describe  how  history  actually  unfolds.  Second,  it  is  impossible  to  count  down  from  negative  infinity  without  encountering the  problem  of  a  potential infinity  that  never  actually  reaches  infinity. For  the  real  world,  therefore,  there  must  be  a  first  event  that  occurred  a  finite  amount  of  time  ago  in  the  past . . . [More] >>

Craig:

>Strictly speaking, I wouldn’t say, as you put it, that a “beginningless causal chain would be (or form) an actually infinite set.” Sets, if they exist, are abstract objects and so should not be identified with the series of events in time. Using what I would regard as the useful fiction of a set, I suppose we could say that the set of past events is an infinite set if the series of past events is beginningless. But I prefer simply to say that if the temporal series of events is beginningless, then the number of past events is infinite or that there has occurred an infinite number of past events . . . .

It might be said that at least there have been past events, and so they can be numbered. But by the same token there will be future events, so why can they not be numbered? Accordingly, one might be tempted to say that in an endless future there will be an actually infinite number of events, just as in a beginningless past there have been an actually infinite number of events. But in a sense that assertion is false; for there never will be an actually infinite number of events, since it is impossible to count to infinity. The only sense in which there will be an infinite number of events is that the series of events will go toward infinity as a limit.

But that is the concept of a potential infinite, not an actual infinite. Here the objectivity of temporal becoming makes itself felt. For as a result of the arrow of time, the series of events later than any arbitrarily selected past event is properly to be regarded as potentially infinite, that is to say, finite but indefinitely increasing toward infinity as a limit. The situation, significantly, is not symmetrical: as we have seen, the series of events earlier than any arbitrarily selected future event cannot properly be regarded as potentially infinite. So when we say that the number of past events is infinite, we mean that prior to today ℵ0 events have elapsed. But when we say that the number of future events is infinite, we do not mean that ℵ0 events will elapse, for that is false. [More]>>

Food for further thought. END

PS: As issues on numbers etc have become a major focus for discussion, HT DS here is a presentation of the overview:

unity

Where also, this continuum result is useful:

unified_continuum

PPS: As a blue vs pink punched paper tape example is used below, cf the real world machines

Punched paper Tape, as used in older computers and numerically controlled machine tools (Courtesy Wiki & Siemens)

Punched paper Tape, as used in older computers and numerically controlled machine tools (Courtesy Wiki & Siemens)

and the abstraction for mathematical operations:

punchtapes_1-1

Note as well a Turing Machine physical model:

Turing_Machine_Model_Davey_2012

and its abstracted operational form for Mathematical analysis:

turing_machine

F/N: HT BA77, let us try to embed a video: XXXX nope, fails XXXX so instead let us instead link the vid page.

1,416 Responses to Durston and Craig on an infinite temporal past . . .

  1. Durston and Craig on a claimed infinite actual past.

  2. Only God has an infinite past if there are even any words to explain where God is, as where He is(in Himself) must be timeless and is not subject to time until He took on human form and only in that sense can it be said God is subject to time because He took time into Himself. We have no linguistics or true comprehension of how this can be, the Kenosis. The emptying of his Deity to become a man. One only has to have faith that the Great I AM is a reality and He is visible in Jesus of Nazareth.

    Part of God’s name He gave Moses “I AM THAT I AM” is part and parcel of Moses’ description of God as “From Everlasting to Everlasting Thou art God.” We simply cannot comprehend this, it is beyond any temporal creature even Angels.

    This is a part of the ontological argument for God that Anselm spoke about in his God as the greatest conceivable Being argument I believe. I am of the opinion that this whole subject and the fact that we can even conceive of a God from Everlasting and speak about Him in any meaningful way shows that we are no mere end product of materialist descent. We came from an Eternal God who in some way imprinted His Image on us. The Imago Dei.

  3. KF,

    I think Craig and I are in agreement, at least regarding a “beginningless” or infinite past. For example:

    So when we say that the number of past events is infinite, we mean that prior to today ℵ0 events have elapsed.

    Durston, however, makes the erroneous assumption that an infinite past implies counting down from infinity to 0, which we have discussed exhaustively with no progress.

  4. Dave @3

    That is why I always bring in the Biblical concepts of time found in God. To me that is the only solution and end of the argument. The old “If something exists now, then something must have always existed.” axiom. I cannot believe I have had people tell me that it is a false statement.

  5. daveS: …which we have discussed exhaustively with no progress.

    Actually, that never happened.

  6. Well, I can tell you it has exhausted me! 😉

  7. DS,

    no step by step completed process of cause-effect [including a counting succession] can but be finite. It will therefore fail to span the transfinite. For many good reasons, reasons reflected in the assertion of those who form natural, counting succession numbers by incrementing, that all naturals are finite. (My concern is that an ordinal succession can be defined and if it is of transfinite scale — endless — not all members can actually be reached by completed counts. As opposed to, we can point out how to reach them in principle.)

    When Craig states “there never will be an actually infinite number of events, since it is impossible to count to infinity” he has elsewhere specifically identified a succession of seconds as relevant events for this. He is not in disagreement — but per the Kalam cosmological is famously in agreement — with Durston when the latter states:

    an infinite real past requires a completed infinity, which is a single object and does not describe how history actually unfolds. Second, it is impossible to count down from negative infinity without encountering the problem of a potential infinity that never actually reaches infinity. For the real world, therefore, there must be a first event that occurred a finite amount of time ago in the past . . .

    But the issue is not so much who agrees/disagrees but what is warranted.

    And, what is warranted is that step by step finite succession cannot bridge to the transfinite. This is easiest to see starting at 0 and counting up, but it is patent that bridging the transfinite the other way to appear at the present has to bridge the same span.

    That is why I went to lengths to identify a reasonable ordered succession

    0, 1, 2 . . . [TRANSFINITE SPAN] . . . w, . . . w + g . . .

    and identify that A = W + g, a transfinite with w the first transfinite ordinal and g some large finite [so still of the scale aleph null] will be such that in a descent

    . . . A, A~1 [= w + (g – 1)], A ~ 2, . . . 2, 1, 0, 1*, 2* . . . n,

    n being now, we see

    A, A~1 [= w + (g – 1)], A ~ 2, . . .

    0, 1#, 2#, . . .

    and so we run into a transfinite bridge and the count down will not reach from A to 0, no more than it can reach up from 0 to A.

    The causal, finite step by step succession of the past will inherently be finite, strongly grounding the conclusion that the past was finite.

    KF

  8. KF,

    As I said, I am just noting that I agree with WLC on the issue of an infinite past. He does not claim anywhere in the quote that it is impossible. Certainly not based on cardinality grounds.

    Regarding your construction, there are no large finite numbers g “of the scale aleph null”. I don’t have to read any further after that error.

    Why don’t you contact WLC and ask him to review your argument?

  9. PS: For further thought, Tim Holt, http://www.philosophyofreligio.....-the-past/ :

    three mathematical arguments for the finitude of the past will be outlined.

    The first argument draws on the idea that actual infinites cannot exist, the second on the idea that actual infinites cannot be created by successive addition, and the third on the idea that actual infinites cannot be traversed.

    If any of these arguments is successful, then the second premise of the kalam arguments will have been proven.

    The Impossibility of an Actual Infinite

    The first mathematical argument for the claim that the universe has a beginning draws on the idea that the existence of an infinite number of anything leads to logical contradictions. If the universe did not have a beginning, then the past would be infinite, i.e. there would be an infinite number of past times. There cannot, however, be an infinite number of anything, and so the past cannot be infinite, and so the universe must have had a beginning.

    Why think that there cannot be an infinite number of anything? There are two types of infinites, potential infinites and actual infinites. Potential infinites are purely conceptual, and clearly both can and do exist. Mathematicians employ the concept of infinity to solve equations. We can imagine things being infinite. Actual infinites, though, arguably, cannot exist. For an actual infinite to exist it is not sufficient that we can imagine an infinite number of things; for an actual infinite to exist there must be an infinite number of things. This, however, leads to certain logical problems.

    The most famous problem that arises from the existence of an actual infinite is the Hilbert’s Hotel paradox. Hilbert’s Hotel is a (hypothetical) hotel with an infinite number of rooms, each of which is occupied by a guest. As there are an infinite number of rooms and an infinite number of guests, every room is occupied; the hotel cannot accommodate another guest. However, if a new guest arrives, then it is possible to free up a room for them by moving the guest in room number 1 to room number 2, and the guest in room number 2 to room number 3, and so on. As for every room n there is a room n + 1, every guest can be moved into a different room, thus leaving room number 1 vacant. The new guest, then, can be accommodated after all. This is clearly paradoxical; it is not possible that a hotel both can and cannot accommodate a new guest. Hilbert’s Hotel, therefore, is not possible.

    A similar paradox arises if the past is infinite. If there exists an infinite past, then if we were to assign a number to each past moment then every [counting] number (i.e. every postive integer) would be assigned to some moment. There would therefore be no unassigned number to be assigned to the present moment as it passes into the past. However, by reassigning the numbers such that moment number one becomes moment number two, and moment number two becomes moment number three, and so on, we could free up moment number one to be assigned to the present. If the past is infinite, therefore, then there both is and is not a free number to be assigned to the present as it passes into the past. [–> notice the link to Hilbert’s Hotel]

    That such a paradox results from the assumption that the past is infinite, it is claimed, demonstrates that it is not possible that that assumption is correct. The past, it seems, cannot be infinite, because it is not possible that there be an infinite number of past moments. If the past cannot be infinite, then the universe must have a beginning. This is the first mathematical argument for the second premise of the kalam cosmological argument.

    The Impossibility of an Actual Infinite created by Successive Addition

    The second mathematical argument for the claim that the universe has a beginning draws on the idea that an actual infinite cannot be created by successive addition. If one begins with a number, and repeatedly adds one to it, one will never arrive at infinity. If one has a heap of sand, and repeatedly adds more sand to it, the heap will never become infinitely large. Taking something finite and repeatedly adding finite quantities to it will never make it infinite. Actual infinites cannot be created by successive addition.

    The past has been created by successive addition. The past continuously grows as one moment after another passes from the future into the present and then into the past. Every moment that is now past was once in the future, but was added to the past by the passage of time.

    If actual infinites cannot be created by successive addition, and the past was created by successive addition, then the past cannot be an actual infinite. The past must be finite, and the universe must therefore have had a beginning. This is the second mathematical argument for the second premise of the kalam cosmological argument.

    The Impossibility of an Actual Infinite that has been Traversed

    The third mathematical argument for the claim that the universe has a beginning draws on the idea that actual infinites cannot be traversed.

    If I were to set out on a journey to an infinitely distant point in space, it would not just take me a long time to get there; rather, I would never get there. No matter how long I had been walking for, a part of the journey would still remain. I would never arrive at my destination. Infinite space cannot be traversed.

    Similarly, if I were to start counting to infinity, it would not just take me a long time to get there; rather, I would never get there. No matter how long I had been counting for, I would still only have counted to a finite number. It is impossible to traverse the infinite set of numbers between zero and infinity. This also applies to the past. If the past were infinite, then it would not just take a long time to the present to arrive; rather, the present would never arrive. No matter how much time had passed, we would still be working through the infinite past. It is impossible to traverse an infinite period of time.

    Clearly, though, the present has arrived, the past has been traversed. The past, therefore, cannot be infinite, but must rather be finite. The universe has a beginning.

    More points to ponder.

  10. DS,

    I suggest you read here http://www.leaderu.com/truth/3truth11.html for an outline of Craig’s Kalam argument and for some elaboration of the following skeleton outline, and notice how Hilbert’s Hotel is applied:

    1. Whatever begins to exist has a cause of its
    existence.

    2. The universe began to exist.

    2.1 Argument based on the impossibility of an
    actual infinite.

    2.11 An actual infinite cannot exist.
    2.12 An infinite temporal regress of
    events is an actual infinite.
    2.13 Therefore, an infinite temporal
    regress of events cannot exist.

    2.2 Argument based on the impossibility of
    the formation of an actual infinite by
    successive addition.

    2.21 A collection formed [–> note tense, denoting completion] by successive
    addition cannot be actually infinite.

    2.22 The temporal series of past events
    is a collection formed by successive
    addition.
    2.23 Therefore, the temporal series of
    past events cannot be actually
    infinite
    .

    ________________

    3. Therefore, the universe has a cause of its
    existence.

    KF

  11. KF,

    It’s not clear to me that he impossibility of the existence of Hilbert’s Hotel implies the impossibility of the existence of an infinite past. “Things” is a pretty general term.

    But, at least WLC’s argument is not based on mathematical misunderstandings, I will say.

  12. PS: Particularly observe:

    Against (2.21), Mackie objects that the argument illicitly assumes an infinitely distant starting point in the past and then pronounces it impossible to travel from that point to today. But there would in an infinite past be no starting point, not even an infinitely distant one. Yet from any given point in the infinite past, there is only a finite distance to the present.[16] Now it seems to me that Mackie’s allegation that the argument presupposes an infinitely distant starting point is entirely groundless. The beginningless character of the series only serves to accentuate the difficulty of its being formed by successive addition. The fact that there is no beginning at all, not even an infinitely distant one, makes the problem more, not less, nettlesome. And the point that from any moment in the infinite past there is only a finite temporal distance to the present may be dismissed as irrelevant. The question is not how any finite portion of the temporal series can be formed, but how the whole infinite series can be formed. If Mackie thinks that because every segment of the series can be formed by successive addition therefore the whole series can be so formed, then he is simply committing the fallacy of composition.

  13. DS, have you actually read the discussion of HH? An infinite number of past seconds occupied in succession by events fits right in. And I have just put up a PS on spanning the transfinite. KF

    PS: Note, my point of concern has always been, that to count up to or down from infinity the transfinite would have to be spanned in step by step finite succession, leading to an impossibility.

  14. KF,

    First notice this:

    And the point that from any moment in the infinite past there is only a finite temporal distance to the present may be dismissed as irrelevant.

    That may be irrelevant to their discussion, but is quite central to ours. WLC agrees with me in that any point in the infinite past is only finitely distant temporally to the present.

    The question is not how any finite portion of the temporal series can be formed, but how the whole infinite series can be formed.

    I assume that a deity who exists outside of time would be able to form such a series. Do you think that even God could not arrange this?

  15. Yes, I read the HH discussion. It presumes a starting point, so it doesn’t parallel my clock example, which has none.

  16. DS, nope he has not done so, but made a statement for sake of argument that shuts out a side discussion that we are now having. I repeat, I am quite concerned that any proposed bridging from a transfinitely remote point — not the beginning of the whole, but a point that is ordinally subsequent (as illustrated) — to the present that suggests that a finite and completed sequence attains the present is questionable or even contradictory. Is the zone bridged transfinite? If yes, no finite sequence of steps can completely bridge it to 0. Is the zone in question one that has been bridged by a FINITE and now completed sequence to now? If yes to this instead, obviously the bridged span is just as FINITE not transfinite. What looks very much like trying to have the cake and eat it too is to propose that a TRANSFINITE span has been bridged by a FINITE, completed step by step descent. KF

  17. The HH example involves a customer attempting to enter the full hotel, so there is a starting point.

    And recall:

    And the point that from any moment in the infinite past there is only a finite temporal distance to the present may be dismissed as irrelevant.

    WLC affirms that “from any moment in the infinite past there is only a finite temporal distance to the present”.

    As I mentioned in the other thread, I’ll step back and let others discuss. If vjtorley is interested in addressing these arguments involving transfinite natural numbers and finite numbers of “order aleph-null”, then I for one would like to see it.

  18. DS, I note to you that if you wish to define “all” integers as finite -which then raises serious concerns on then claiming the cardinality of the set of integers is transfinite if such be applied — a finite integer must needs be a finite span of “units” or steps from 0 — that does not remove from the table the span of ordinal whole numbers suitable for counting sequences. That is why I have latterly stayed strictly away from appealing to “natural numbers” and “integers.” I point out that what holds a finite value cannot at the same time be a transfinite span from the origin, 0. Likewise, no finite span of steps will be able to traverse the transfinite span. The challenge remains, to traverse the transfinite in a stepwise process that rests on finite stages. Assertions about at a given time it stands done do not answer the issue. Hence my use of A above. KF

    PS: For HH, simply revert to negatively numbered rooms, so the regression is open ended in the opposite to usual direction: . . . -2, -1, 0. Each room is now a second, say, and the tenant in it the events thereof. Has anything fundamental about the problems of actualising such a hotel changed? Try, the manager inspects each room in turn, and has been doing so forever at a rate of one per second. When does he arrive at the front desk, 0? Or imagine, builders are building the rooms at one per day forever, when will they get to building the front desk? (Can they reach room 0 from room w + g, aka room A, which is transfinitely remote from 0, given that a day by day sequence of room builds is inherently finite? Does building A, A~1, A~2, . . . in sequence [an inherently finite process] arrive at 0 by traversing the transfinite distance to 0? Where the span to 0 is endless?)

  19. KF,

    I point out that what holds a finite value cannot at the same time be a transfinite span from the origin, 0.

    Yes, that’s something I’ve been saying all along.

    Again, put all this together and run it by vjtorley, please.

    Edit: Re: your HH explanation: If the manager was in room number -100 one hundred seconds ago, he arrives at the desk now.

  20. DS, do you not see that a transfinite span to 0 then runs into a problem when it is to be spanned by an inherently finite process? KF

  21. DS, Yes a manager can span the finite in finite time. But the issue is to span the proposed transfinite with an inherently finite stepwise process. KF

  22. KF,

    Gotta run. In the scenario I described above, the manager was in room -n n seconds ago, for each natural number n.

    Given any room in the hotel, I can tell you when he was there.

  23. DS, being in room n, n seconds past does not bridge to reaching the front desk at 0 when we deal with the transfinitely remote rooms; when also the inspection process is a finite step by step process. And that does not get into the problems of managing the suggested actual hotel as a whole that are independent of his inspection tour. Assume he has a smart phone and a PA system that reaches every room, with an assistant at front desk. Infinitely many new guests arrive and he is full but by asking guests in room -n to go to room 2n suddenly he can put the new ones into 2n – 1, and yet will have the same number of guests after such a move. And so forth. KF.

  24. KF,

    PA system?? Smart phone?? We don’t need to get into that. Obviously this hotel is not physically realizable, but that’s not the point.

    The manager was in room -n n seconds ago, for all natural numbers n.

    He completed the inspection of every single room just now.

    I don’t know if that’s what you call “spanning the transfinite”, but it’s quite parallel to the clock example.

    You can’t show it’s impossible just based on cardinalities.

  25. DS, how do you span the transfinite with the inherently finite? KF

  26. KF,

    I have simply been saying that a clock ticking throughout an infinite past up to the present cannot be defeated using simple cardinal arithmetic. That’s really in essence identical to this Hilbert Hotel example, which I have just explained.

    I don’t know if that counts as “spanning the transfinite with the inherently finite”, but it’s all I have been asserting.

  27. KF, I know my thoughts on this are an oversimplification but it makes sense math-wise, and might contain a clue for you to work on.

    Energy to achieve maximum Radius of universe is mathematically possible from nothing (zero) by separating the nothingness into equal positive and negative halves:

    0=R-R
    or:
    -R=0-R
    or:
    +R=0–R = +R=0+R

    The phase Angle (one Radian or 0 to 360 degrees each) is analogous to time but only repeats itself in cycles, as opposed to a continuous timeline.

    In 1D is a sine wave:
    X=Cos(A)*R

    Adding Y for 2D makes a circle:
    Y=Sin(A)*R

    Adding Z for 3D makes a sphere. Calculate X,Y,Z using your favorite spherical coordinate formula or matrix math.

    The law of conservation of energy is not violated. Only thing required is that nothingness cannot exist without some wave forming imbalance that we in turn perceive as a universe containing matter and antimatter.

  28. Gary writes,

    0=R-R
    or:
    -R=0-R
    or:
    +R=0–R

    Uh, no. The part after the second “or” does not follow. You just went from -R to +R with no justification at all. I have no idea what the point is, but that is wrong.

  29. daveS

    At this point in the debate with KF, you have had many people ( some with advanced degrees in math ) try to explain this to you. It is clear you don’t understand the concept.

    It may be an assumption of mine, but I assume you are also a naturalist. You are not helping your cause by showing how dogmatically you can hold a position, even when others can see that your position is not true.

  30. JDH,

    Would you please point out my mathematical errors then?

  31. Lessee, what would it be like to live in a universe that’s infinitely old?

    – Due to inflation, many stars would be infinitely far apart

    – There would be an infinite number of dead stars filling up space, blocking out the light from new stars

    – There would have to be a way that new stars would spontaneously appear, and at a rate consistent with cosmic inflation

    – The entropy of the universe would be maximized, just below the rate of new stars magically appearing from somewhere

    – Everything that could have happened happened already

    – Science would finalize its divorce from reality, and evolve into a system of whose sole purpose is to rationalize observations into a philosophically acceptable narrative for atheists

    – The Multiverse hypothesis could be abandoned as redundant

    – A lot of people would have to apologize for arguing that if God exists, who created God. It’s turtles, gods, and stars all the way down.

    -Q

  32. Uh, no. The part after the second “or” does not follow. You just went from -R to +R with no justification at all.

    I admit to not being a math-wiz. And creating two opposites of something from nothing using the equation 0=R-R where (R is not equal to 0) is in a way oversimplifying. But then again in works in math so who knows?

    If you are good at showing all the possible ways to substitute the equation variables then please show me what you find. At least in computer logic it’s the start of a very simple oscillation that (with no forever increasing Time variable) would go forever into infinity, if it were not for power blackouts and computers not being expected to be able to stay going that long.

  33. Infinity is a religion of cretins. Just saying.

  34. Let’s see what a physicist at the University of Nottingham does with an infinite number series . . .

    https://www.youtube.com/watch?v=w-I6XTVZXww

    Then, as you can clearly see from the math (which is used in string theory) that if we assign a billion years for each number, the sum allows us to move backward in time! o.O

    -Q

  35. Querius:

    the sum allows us to move backward in time

    Motion in time is an absurdity. It’s not even wrong.

  36. Mapou, we are dependent on a past causal succession that has led to the present. While time is evidently unidirectional and so far as we see open ended to the future, we may and do profitably wish to ponder that past by moving backwards in analytical thought. Not least as the weight of that cumulative past informs us about underlying regularities of behaviour, trends and values of key parameters that can guide us in acting now to favourably shape the future. KF

  37. GG, nothingness in this context means non-being. Not matter, not energy, not time, not laws and forces, not mind. We may indeed profitably discuss levels in potential field and conservation but energy in physics is never in itself a negative. As one clue notice how kinetic energy is associated with a velocity squared term, the mass-energy relationship depends on speed of light squared, and in the force-displacement dot product formulation dW = F dot dx work signs do not split into something from nothing but into energy flows from one type to another. Ponder in this context energy conservation. KF

    Algebra: 0 = R – R, rearranged R = R. This is little more than the RHS restates the same value and kind as the LHS. This does not flash such into physical actuality. An equals sign has no power to create. Especially, something from nothing. (Perhaps, inadvertently, we should see that it is a thought, pointing to a mind reflecting on logical-quantitative and structural relationships. A mind is something creative and in theism God is the ultimate creative Mind!)

  38. DS, the core issue seems to be conceptual. If we have had an infinite step by step temporally manifested causal succession to now, it has had to span a transfinite domain to reach the present or any finitely remote past point you care to identify. This is problematic given that any step k is succeeded by k + 1, etc, leading to an inherently finite successive pattern of cumulative steps — something you and the sources you cite appeal to in claiming that all natural numbers are finite.* Substituting inherently finite spans of discrete cumulative steps across the finite past does not succeed in replacing or removing that challenge. You have acknowledged the inherent finitude of stepwise processes and thus by implication the impotence of same to span the transfinite range that is required. In short, the processes you accept do not have the power to address the challenge in hand, spanning the transfinite. KF

    PS: Hilbert’s Hotel shows, spectacularly, why the abstract infinite would be utterly infeasible in the physical world. A full hotel [which means every room is occupied] that by reassigning rooms suddenly can hold a further infinity of guests? A manager able to inspect all the rooms in finite stepwise succession down to the reception hall and desk? [We call that room 0.] Workers able to build and complete it room by room? Where, physical feasibility is a constraint on physical actuality.

    * I agree that any number we actually reach or exceed by step by step counting is finite, but am concerned that the span of counting numbers is in principle unlimited in range thus transfinite. The ellipsis — . . . — may inadvertently disguise that a transfinite range thus a potential transfinite span rather than an actually traversed one, has been put into the discussion. The ordinals gives us a way to discuss this, and that is what I used above and previously. 1/m = A where A is also w + g, beyond the first transfinite ordinal and is of cardinality aleph null arises in that context as a way to put in symbolic terms the issue of concern.

  39. to Q at 33. I don’t believe anyone in this discussion is arguing that the universe has existed for an infinite amount of time. They are arguing about the abstract nature of time as it might be stretching back before the start of the universe.

    If I’m wrong about this, kf and ds, please correct me – you haven’t been discussing time within our universe, have you?

  40. KF,

    Could you translate your mathematical objection about “spanning the transfinite with the inherently finite” to the Hilbert Hotel? I’ve described very simply how the manager could inspect each of the infinitely many rooms of the hotel, with each step simply moving to the next room over. What’s the mathematical issue?

    Regarding the physical feasibility of the HH, of course that’s true. The point is it’s a concrete model which allows us to reason about the abstract more easily.

    You could make the hotel less spectacularly infeasible by requiring it to have only two rooms, with the manager going back and forth between rooms once per second, with this process having continued into an infinite past. I don’t think it’s been shown that a physical analog of this process is impossible (see the oscillating universe models which remain unrefuted at this point).

    But again, I’m interested in the mathematics here, not the physics.

  41. Aleta @41,

    Yes, just to clarify, I think it’s very unlikely that our universe is infinitely old, not that my opinion means anything. I’m not trying to prove it is.

    I’m simply interested in analyzing KF’s argument against an infinite past. While I don’t think it succeeds, at least it’s new (to me).

    Ultimately, I suppose we are discussing time in the universe, but we have been mostly focusing on the properties of number systems, the infinite, etc., as you said.

  42. Aleta, the argument on the table — in now a third current thread — is in fact that the universe (taking in multiverse proposals etc) is infinitely old. Mathematical considerations are tied to the issue of there being a proposed endless causal stage by stage succession of events to the present. However, inherently a step by step countable process is inherently finite if completed. The span to be addressed is by contrast transfinite. KF

  43. DS, you have described how the manager could inspect finitely many rooms in an inherently finite process. The problem is the span in question is transfinite and endless in the first degree, of scale aleph null. KF

    PS: You have now added oh try two rooms and infinitely many cycles between the two. This still leaves on the table the issue of finiteness of stepwise process.

  44. KF,

    No, I have described how the manager could inspect every room in the Hilbert Hotel.

    If you disagree, name a room that he missed, or otherwise prove that one exists.

    Please note that every room number is finite.

    Edit:

    PS: You have now added oh try two rooms and infinitely many cycles between the two. This still leaves on the table the issue of finiteness of stepwise process.

    Obviously. That’s the point.

  45. KF:

    While time is evidently unidirectional

    This, too, is an illusion created by the way the way our brain stores memory as a sequence that can be scanned as such by our consciousness. IOW, it’s not even wrong. The idea of a unidirectional arrow of time is pure nonsense. There is only the present, the NOW.

    I have explained many times on UD why there can be no such thing as changing time or motion in time. It is a self-referential fallacy. I would do it again but I’m afraid it will fall on deaf ears. So this is my last post in this thread. Have fun with your misconceptions.

  46. There is only the present, the NOW.

    But there are relational changes amongst the objects of the universe. And some of these objects have memory too, besides brains, such as computers, and decks of cards, that record the uni-directionality of these state changes.

    For example, take a BlackJack game. The deck of cards going from one relational state progressively to another. The odds of the game change due to these relational state changes. The system has memory, and it is based on a “direction” of the state changes.

    The flow of these state changes is called “time.” Even though consciousness is always in the “now”, consciousness can change its state as a result of the perception of the progressing state changes among objects. Close your eyes. Consciousness sees black. Open your eyes. Consciousness sees colors.

  47. Mapou, time is causally connected. KF

  48. DS, you have yet to tell us how a finite step by step process traverses the transfinite span of an actual infinity. KF

  49. KF,

    I have told you how the manager traverses the Hilbert Hotel in a step-by-step fashion. That’s analogous to my infinite clock example: The manager carries a pocket watch which ticks once per each room he inspects. That’s what I said I could do.

  50. DS, nope; the issue of stepwise traversal of a transfinite span remains. KF

  51. KF,

    You’re certainly welcome to show that the manager missed a room under my scheme.

  52. DS, you have acknowledged that a step by step process is inherently finite (even citing with approval arguments based on it). The task in hand is to span the transfinite, and at some point the manager would have been transfinitely far from the last room no 1 and the front desk room 0. Step by step successive descent at that level will never bridge down to a finite range from 0 as moving a finite set of steps away from a transfinite point say room w + g to w + (g – 1), w + (g – 2) etc will still be of the same cardinality aleph null away from the front desk; w being the first transfinite ordinal omega which holds cardinality aleph null. Counting up from 0 in steps will never surpass finite values and likewise trying the same span the other way around will not be any more successful. As, has been pointed out to you as a concern any number of times now. Notice, too, when the new guests come the message is broadcast so changes in rooms are all at once [old guests previously in rooms n to rooms 2n, new ones to 2n-1], i.e. in parallel. That is a clue on how action in successive individual steps would fail. KF

    PS: Notice how Robinson et al have worked in nonstandard analysis to bridge to the transfinite. They use the power of the hyperbolic function y = 1/x for x –> 0 to move in scale from the v small to the very large. Then put in an infinitesimal, to result with a hyper-real, sometimes a hyper-integer. I suggest a more modest “catapult” from some convenient infinitesimal m to y = 1/m = A = w + g, this being a transfinite whole number value of cardinality aleph null. Then, as the manager descends in order, at some juncture he inspects room A.

  53. M62, Your card game example aptly shows the cumulative, causal transition of stages with time and the unidirectional flow. A fresh deck in order is disordered by shuffling and going back spontaneously is a rare event indeed. This reflects the point of trend to increased entropy through moving to config clusters of higher statistical weight. KF

  54. KF,

    DS, you have acknowledged that a step by step process is inherently finite (even citing with approval arguments based on it). The task in hand is to span the transfinite, and at some point the manager would have been transfinitely far from the last room no 1 and the front desk room 0.

    I was just at the gym thinking that at least we are not talking about rooms infinitely far from the front desk anymore. Oh well.

    At no point was the manager at some transfinite distance from room 0. Every room number is finite, hence every room has finite distance to the front desk.

    Edit: In response to your PS, I am working with a Hilbert Hotel with room numbers equal to the opposites of the natural numbers. No transfinite room numbers allowed.

  55. nm

  56. Let me explain why all this hyperreal stuff is moot.

    The issue is whether the past could be infinite. If the past was infinite, then that would certainly mean that given any natural number n, the universe already existed n seconds ago. IOW, -n seconds is a valid time coordinate for our universe (assuming the origin is set at the present).

    This does not necessarily mean that given some infinite hyperreal integer A, then the universe already existed A seconds ago.

    I am not supposing such. If you present me with an infinite hyperreal integer A, I don’t know if -A is a valid time coordinate for our universe.

    That’s why you have to stick to arguing against the Hilbert Hotel example where the room numbers all come from N (or their opposites), and where each room is finitely many steps from the front desk.

  57. DS, have you noticed the lead ellipsis in the series? That means the issue is onwards from A, for the sake of the argument. Note 7 above:

    what is warranted is that step by step finite succession cannot bridge to the transfinite. This is easiest to see starting at 0 and counting up, but it is patent that bridging the transfinite the other way to appear at the present has to bridge the same span.

    That is why I went to lengths to identify a reasonable ordered succession

    0, 1, 2 . . . [TRANSFINITE SPAN] . . . w, . . . w + g . . .

    and identify that A = W + g, a transfinite with w the first transfinite ordinal and g some large finite [so still of the scale aleph null] will be such that in a descent

    . . . A, A~1 [= w + (g – 1)], A ~ 2, . . . 2, 1, 0, 1*, 2* . . . n,

    n being now, we see

    A, A~1 [= w + (g – 1)], A ~ 2, . . .

    0, 1#, 2#, . . .

    and so we run into a transfinite bridge and the count down will not reach from A to 0, no more than it can reach up from 0 to A.

    The causal, finite step by step succession of the past will inherently be finite, strongly grounding the conclusion that the past was finite.

    If you cannot get a tansfinite span by incrementa steps after A, it matters not for the argument at this point what may lie beyond A.

    But then the same logic applies a second time and there is no indefinitely transfinite preceding span for any value. That is the sequence is finite, the past is finite.

    Time began a finite span ago.

    KF

    PS: I used the multiplicative inverse of m and linked mention of the hyper-reals to show how one can get to a transfinite. All it needs is recognising that the interval [0, 1] is continuous, and so there are values arbitrarily, infinitesimally close to 0. Closer than any epsilon neighbourhood of 0.

    PPS: Without room numbers in the ordered, numbered sequence of rooms that are of scale w [standing in symbolically for omega], of aleph null cardinality, there will not be an actual infinity of rooms. There is a difference between an unspecified large but finite value and an actual infinity, which is what the hotel is supposed to be.

  58. KF,

    Your quoted construction is gibberish, I’m sorry to say.

    Regarding your PPS, the room numbers are 0, -1, -2, -3, … .

    There are infinitely many rooms, but each room number is finite. Please, no more ill-defined expressions such as “of scale little-omega”.

    There are so many mathematicians involved with ID. Dembski, Sewell, Berlinski, probably quite a few posters here. Have you ever run any of this stuff by any of them?

  59. GG, nothingness in this context means non-being. Not matter, not energy, not time, not laws and forces, not mind.

    KF, in a computer model like I described all of that including time(steps) have to somehow be coded into it or else absolutely nothing ever exists in the virtual world. The methodology forces everyone to start with a total nothingness (i.e. dimension an empty array to put things like forces into) then supply the coded math/logic that makes a world as close as possible to ours form inside.

  60. Aleta, you might want to investigate why, starting with Einstein, physicists refer to something that they call space-time.

    Gary, the problem that a lot of people have is that they don’t understand nothing. Nothing in this context is non-existence. For example, the Easter bunny is non-existent. So, imagine how a universe spawns out of the Easter bunny.

    -Q

  61. DS,

    I notice:

    There are infinitely many rooms, but each room number is finite. Please, no more ill-defined expressions such as “of scale little-omega”.

    DS, the highlighted tells a long tale.

    When a set is presented as “the room numbers are 0, -1, -2, -3, … “ that ellipsis is very important, pointing to the transfinite character of the set. This means that at some scale one is transfinitely far away from the beginning. That is what gives the problem of completing the stepping down process from the endlessly high value zone to the finite neighbourhood of 0.

    As for “gibberish,” I again point to what I noted at 56 in the previous thread:

    Now, the count and successive establishment of counting numbers from {} –> 0, to {0} –> 1, to {0,1} –> 2 etc suggests looking at ordinal numbers as an approach. And such is obviously foundational.

    Where, for convenience let us refer Wiki (which in this context from my POV is inclined to be seen as testifying against its ideological interests) . . . and where I use w for omega and E for epsilon:

    In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by labelling the objects with distinct whole numbers. Ordinal numbers are thus the “labels” needed to arrange infinite collections of objects in order. Ordinals are distinct from cardinal numbers, which are useful for saying how many objects are in a collection. Although the distinction between ordinals and cardinals is not always apparent in finite sets (one can go from one to the other just by counting labels), different infinite ordinals can describe the same cardinal (see Hilbert’s grand hotel).

    Ordinals were introduced by Georg Cantor in 1883[1] to accommodate infinite sequences and to classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[2]

    Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is w [–> omega], which is identified with the cardinal number aleph_0. However, in the transfinite case, beyond w, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely aleph_0 itself, there are uncountably many countably infinite ordinals, namely

    w, w + 1, w + 2, …, w·2, w·2 + 1, …, w^2, …, w^3, …, w^w, …, w^[w^w], …, E_0, ….

    Here addition and multiplication are not commutative: in particular 1 + w is w rather than w + 1 and likewise, 2·w is w rather than w·2. The set of all countable ordinals constitutes the first uncountable ordinal w_1, which is identified with the cardinal aleph_1 (next cardinal after aleph_0). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals to cardinals . . .

    This at least looks promising, as it clearly points to whole numbers of transfinite nature, and distinctly identifies increments by addition to the next ordinal. Where cardinality at transfinite scale is an index of order of magnitude expressed at aleph null level by one to one correspondence.

    The logical next step is to suggest some finite counting number g, to be added to w, and put up as a further construction of A:

    1/m = A

    (That is A * m = 1, multiplicative inverse. Where, m is an infinitesimal.)

    A = w + g

    In this context A less 1 would be w + (g – 1) . . . let us symbolise as A ~ 1, and so forth.

    Under these circumstances, it seems to me for the moment that A would be a transfinite not actually reachable from 0 by an inherently finite step by step process but is a whole number in an identifiable sequence.

    Reversing the matter let us now look at:

    . . . A, A ~ 1, A ~ 2, . . . 2, 1, 0, 1*, 2*, . . . n*

    A is obviously not a first step, the leading ellipsis takes care of that. For all we know for the moment an indefinitely large descending sequence has arrived at A. At least, we must be open to it.

    But, now we go beyond A and can make a correspondence of onward steps trying to descend to 0, say to be tagged with the singularity:

    A, A ~ 1, A ~ 2, . . . 2, 1, 0 . . . n*

    0#, 1#, 2# . . .

    We face an inherently finite state based descent that can only ever be completed to a finite extent. But the span to be traversed to 0 is transfinite.

    Let me augment the list of ordinals of scale aleph null:

    w, w + 1, w + 2, . . . w + g [g a large finite value], …, w·2, w·2 + 1, …, w^2, …, w^3, …, w^w, …, w^[w^w], …, E_0, ….

    where of course, ordinals will go:

    {} –> 0, to {0} –> 1, to {0,1} –> 2, . . . k, . . .

    [a transfinite span –> let’s add: for any specific finite k you please you can count on forever in here, k+1, k + 2 . . . and put the onward count in correspondence to 0, 1, 2 . . . or perhaps better 1, 3, 5 . . . having already put everything so far in correspondence with 2, 4, 6 . . . as 1 can here be identified by its being odd as strict successor to all evens (notice, use of the definition of a transfinite set as enfolding an endless strict subset) ]

    . . . w, w + 1, w + 2, . . . [as above]

    with reminder:

    The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is w [–> omega], which is identified with the cardinal number aleph_0. However, in the transfinite case, beyond w, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely aleph_0 itself, there are uncountably many countably infinite ordinals

    My concern has been that for the finite counting numbers laid out in ordered sequence, 0, 1, 2 . . . , the cardinality of successive ranges will equal the value of the last listed member. So, a stepwise succession to k will hold cardinality k and this is then exceeded by k + 1 of cardinality k + 1. But in every case, the cardinality is therefore just as described, finite. As in not transfinite, not endless, not of order aleph null.

    But, we then see that the set of numbers — including of room numbers [how can they be put on the doors?] — we actually require needs to be of transfinite character, endless. That requires going to transfinite cardinals of order w.

    The room inspection tour must start in the transfinite zone and arrive at the finite one in order to inspect all rooms. This cannot be completed, no more than it can be completed to increment in steps to the transfinite while being finitely far from the start at each successive, cumulative step.

    In short, my concern is that the ellipsis does a lot of work, and may be implicitly covering over a supertask not feasible of completion.

    KF

    PS: The captcha pops up and goes white screen again.

  62. F/N: Wiki on the hotel:

    Hilbert’s paradox of the Grand Hotel is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and that this process may be repeated infinitely often . . . .

    The paradox

    Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms, where the pigeonhole principle would apply.
    Finitely many new guests

    Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests.
    Infinitely many new guests

    It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n, and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.
    Infinitely many coaches with infinitely many guests each
    For more details on this topic, see Pairing function.

    It is possible to accommodate countably infinitely many coachloads of countably infinite passengers each, by several different methods. Most methods depend on the seats in the coaches being already numbered (alternatively, the hotel manager must have the axiom of countable choice at his or her disposal). In general any pairing function can be used to solve this problem. For each of these methods, consider a passenger’s seat number on a coach to be n, and their coach number to be c, and the numbers n and c are then fed into the two arguments of the pairing function . . . .

    Analysis

    Hilbert’s paradox is a veridical paradox, it leads to a counter-intuitive result that is provably true. The statements “there is a guest to every room” and “no more guests can be accommodated” are not equivalent when there are infinitely many rooms. An analogous situation is presented in Cantor’s diagonal proof.[3]

    Initially, this state of affairs might seem to be counter-intuitive. The properties of “infinite collections of things” are quite different from those of “finite collections of things”. The paradox of Hilbert’s Grand Hotel can be understood by using Cantor’s theory of transfinite numbers. Thus, while in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert’s aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total “number” of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is aleph_0.

    Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: There is a bijection from the naturals to the rationals.

    This of course raises many issues.

    The point being, the power of endlessness.

    Which requires transfinite character.

    Where also the situation of having an ordinal immediate successor or immediate predecessor does not mean that a number w + r, r finite, cannot be of transfinite cardinality.

    From this we see that an inherently finite stepwise cumulative process of in effect counting causal succession will not exhaust the transfinite span. If we hold 0, 1, 2, . . . k, k+1 to be inherently finite to k, the span will not be transfinite. Countable in principle does not mean that one can complete an endless counting process in praxis. As the very word “endless” suggests.

    If it is thus a supertask to attempt to count up endlessly to arrive at w etc of order of magnitude aleph null, by the same logic it is equally a supertask to try to count down from that scale across an endless span to a finite neighbourhood of 0, of “radius” n.

    In short there is a difference between in principle and in praxis.

    Much lurks beneath the ellipsis and we must be very clear as to whether it speaks of a finite and complete-able process or an endless span that cannot be completed by a finite succession of finite steps.

    Hence the significance of a more powerful process 1/m = A.

    KF

  63. Q & GG: Or, ponder how a universe spawns from the dreams of a rock [vs Divine, eternal contemplation], following Aristotle . . . on recognising that rocks have no dreams. Nothing, properly denoting non-being. KF

  64. GG, of course you are familiar with the von Neumann type construction which starts at the empty set {} and then goes as I outlined at 63: {} –> 0, to {0} –> 1, to {0,1} –> 2, . . . k, . . . , thus getting to all whole numbers — in principle. You may find my discussion here on (fairly lengthy because of many stages of issues; pick up after the pic and discussion of the flying spaghetti monster . . . ) how this can move to a virtual world interesting. Then, if you are a mind of adequate cosmos-forming power it’s fiat lux and poof, we are in the province of a world of space-time, temporal-causal succession. KF

  65. KF,

    DS, the highlighted tells a long tale.

    When a set is presented as “the room numbers are 0, -1, -2, -3, … “ that ellipsis is very important, pointing to the transfinite character of the set. This means that at some scale one is transfinitely far away from the beginning.

    Please prove this, including a rigorous definition of the phrase “at some scale”.

    Things you can’t do:

    * Use “real infinitesimals”, which do not exist.

    * Use hyperreal infinitesimals, because their reciprocals are not real numbers. All the numbers in the set {0, -1, -2, -3, … } are real.

  66. DS

    Please notice what I did at 63, in context including:

    Let me augment the list of ordinals of scale aleph null:

    w, w + 1, w + 2, . . . w + g [g a large finite value], …, w·2, w·2 + 1, …, w^2, …, w^3, …, w^w, …, w^[w^w], …, E_0, ….

    where of course, ordinals will go:

    {} –> 0, to {0} –> 1, to {0,1} –> 2, . . . k, . . .

    [a transfinite span –> let’s add: for any specific finite k you please you can count on forever in here, k+1, k + 2 . . . and put the onward count in correspondence to 0, 1, 2 . . . or perhaps better 1, 3, 5 . . . having already put everything so far in correspondence with 2, 4, 6 . . . as 1 can here be identified by its being odd as strict successor to all evens (notice, use of the definition of a transfinite set as enfolding an endless strict subset) ]

    . . . w, w + 1, w + 2, . . . [as above]

    with reminder:

    The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is w [–> omega], which is identified with the cardinal number aleph_0. However, in the transfinite case, beyond w, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely aleph_0 itself, there are uncountably many countably infinite ordinals

    My concern has been that for the finite counting numbers laid out in ordered sequence, 0, 1, 2 . . . , the cardinality of successive ranges will equal the value of the last listed member. So, a stepwise succession to k will hold cardinality k and this is then exceeded by k + 1 of cardinality k + 1. But in every case, the cardinality is therefore just as described, finite. As in not transfinite, not endless, not of order aleph null.

    KF

  67. KF,

    There is no proof there of the existence of any transfinite numbers in the set {0, -1, -2, …}.

  68. Not only is it logically impossible to count back an infinite amount past number of natural events prior to this time which shows that we could never have reached this moment in time if there were an infinite amount of past natural events, but the idea of “infinite finiteness” which the materialist would have to believe in, if he holds to never ending finite natural events, is a tautological oxymoron.

    The Materialist would have to dump logic on both counts in order to hold to that particular idea.

  69. to DaveS: transfinite numbers are not real numbers (or natural numbers). Transfinite numbers, which may be a misleading name, are “numbers” which represent different levels of infinity. The number of natural numbers, which is infinite, is called aleph null. This is the smallest order of infinity. The number of real numbers, aleph one, is also infinite, but a greater order of infinity: there are more reals than natural numbers. These are not numbers in the same sense that the reals are numbers.

    So Dave is correct when he says “There [are no] transfinite numbers in the set {0, -1, -2, …}. Saying, perhaps, that there is the transfinite number alpha null in the set because the set goes on forever, would be wrong: that would be mixing apples and oranges. The number of numbers in the set is aleph null, but aleph null is not in this set.

    Is this clear? Does kf agree, or not?

  70. JJ:
    Not only is it logically impossible to count back an infinite amount past number of natural events prior to this time

    Seems reasonable, I thought KF’s argument was nothing can happen at all if time was infinite, that infinite time was like walking the wrong way on a moving sidewalk, every time you take a step you are further from your destination, the present.

    which shows that we could never have reached this moment in time if there were an infinite amount of past natural events,

    When an event occurs it happens in its present, an infinite number of events equal an infinite number of presents

    but the idea of “infinite finiteness” which the materialist would have to believe in, if he holds to never ending finite natural events, is a tautological oxymoron.

    An infinite set can be broken up into a infinite number of finite sets.

  71. Aleta,

    to DaveS: transfinite numbers are not real numbers (or natural numbers). Transfinite numbers, which may be a misleading name, are “numbers” which represent different levels of infinity. The number of natural numbers, which is infinite, is called aleph null. This is the smallest order of infinity. The number of real numbers, aleph one, is also infinite, but a greater order of infinity: there are more reals than natural numbers. These are not numbers in the same sense that the reals are numbers.

    Yes, I agree. KF claims to the contrary that the set {0, -1, -2, … } does have transfinite elements. He has stated that if every element of {0, -1, -2, … } were finite, then that set would have finite cardinality.

  72. JJ, The issue is, what does it mean to suggest an infinite descent in finite causal steps from an infinite past. And as I am about to discuss further, I have some serious points of concern with what is being given to us. KF

  73. Aleta,

    I would add that some of the transfinite/infinite numbers KF refers to are hyperreals.

  74. Hi kf. I agree with you about not being to get here from an infinite past. But I am confused about some other issues under discussion. Would you be able to say you agree or disagree with the following statements, from a purely mathematical point of view.

    A: every element of the set of natural numbers N = {0, 1, 2, 3, … } is a finite number.

    B: the number of of numbers in the set N is a transfinite number aleph null.

  75. Aleta

    I have been expressing some concerns and — given the way there has been back-forth for some time — it looks like I will need to go to start-points.

    So, to the construction of (basic sense) counting numbers:

    {} –> 0, to {0} –> 1, to {0,1} –> 2, . . . k [some finite value], . . .

    The ellipsis shows, continue to arbitrary length and it keeps going.

    Now, let us go to what I will for convenience call counting sets:

    {1}, of cardinality 1

    {1, 2} of cardinality 2

    . . .

    {1, 2, . . . k} of cardinality k.

    The idea being, that when something is to be counted, it can be matched 1:1 perfectly to the appropriate counting set.

    In general for k finite, the cardinality will implicate the presence of a member of that position, kth, in the stepwise sequence.

    This may then be exceeded by k + 1, by taking a succeeding step.

    This goes somewhere interesting and to where my concerns lie:

    I note how k is a finite value, and there is an onward ellipsis, such that we then see an exceeding successor, k + 1, then:

    {} –> 0, then {0} –> 1, then {0,1} –> 2, . . . k, . . .

    [a transfinite or endless onward span –> let’s add: for any specific finite k you please you can count on forever in here, k+1, k + 2 . . . and put the onward count in correspondence to 0, 1, 2 . . . or perhaps better 1, 3, 5 . . . having already put everything so far in correspondence with 2, 4, 6 . . . as 1 can here be identified by its being odd as strict successor to all evens (notice, use of the definition of a transfinite set as enfolding an endless strict subset) ]

    . . .

    So, we see here that the endless version of the ordered counting numbers, {1, 2, 3, . . . } can be such that we put a proper subset in one to one match with it with both being endless.

    So, to claim that the set is endless and thus of transfinite cardinanlity aleph null — notice, I am NOT saying that aleph null is in the set somehow — seems to be in direct irreconcilable conflict with claims that all its members are finite, when they are labelled the natural numbers.

    Be that as it may, I step aside from the matter. As my real interest lies in descent from the transfinite.

    We have been dealing with ordered numbers, with a first member and distinct succession, with a ranking/succession rule that is strictly applicable, continuing endlessly.

    So, let us deal with ordinal numbers as ordinal numbers in succession from 0 or 1 as first depending on interest. I usually start with 0.

    Now, let us go to such a succession that goes on through ellipsis to the zone where we pick up w and its successors, w being omega the number that is the “first” ordinal of cardinality aleph null:

    {} –> 0, to {0} –> 1, to {0,1} –> 2, . . . k, . . .

    [a transfinite span, as above with counting numbers in succession]

    . . . w, w + 1, w + 2, . . .

    As, from w there is an onward succession:

    w, w + 1, w + 2, …, w·2, w·2 + 1, …, w^2, …, w^3, …, w^w, …, w^[w^w], …, E_0, ….

    I then believe I may next reasonably identify one such successor to w that is of interest, w + g , which will be of the same cardinality aleph null, where I am viewing aleph null as in effect an index of order of scale being countably transfinite, with its successors being of higher order of scale and not being countably transfinite; whether the order of scale of the continuum c belongs to the sequence is of course notoriously undecidable:

    w, w + 1, w + 2, . . . w + g [g a large finite value], …, w·2, w·2 + 1, …, w^2, …, w^3, …, w^w, …, w^[w^w], …, E_0, ….

    Let us call w + g, A, where also given that [0, 1] is continuous, some infinitesimal m will be its multiplicative inverse: A = 1/m, similar — note I claim no more than comparability — to how the nonstandard analysis comes into play elsewhere.

    Now, given the endless succession from k on, I suggest that in up-counting from a given finite point in the span of numbers, k, we will never reach w in successive finite incremental steps.

    Likewise, it seems, we may reasonably list in reverse form:

    . . . A, A less 1 [i.e. W + (g-1)],. . . w, . . . 2, 1, 0

    Thus — without assuming that there is anything but an indefinitely large span beyond A to the left — we can look at down-counting from A, symbolising [w + (g – 1) as A ~1, etc:

    A, A ~ 1, A ~ 2, . . . w + 1, w, . . . r, . . . 2, 1, 0

    0#, 1#, 2# . . .

    Now, g is finite but very large, this allows us to establish an ordering down to w.

    My key concern is that the ellipsis beyond w is endless, so the stepwise down count sequence that begins at A will go on forever without bridging it to r, a finite neighbourhood of 0.

    Indeed, it seems that once the secondary count started at A hits w at g steps later, and goes on, we are in the position of trying to count across an endless span.

    Thus, a stepwise count process will not reach r, much less the interval [0,1].

    To me, this seems to give some substance to the remark that an inherently finite stepwise counting process will not bridge a transfinite span, will not traverse a transfinite range.

    This is as distinct from that we may set in order a succession that as a set will define a transfinite span of ordered succession. Countable in principle, inexhaustible in practice. The gap between the potentially infinite process and the actually completed infinite process.

    To span the transfinite, it seems to me we need the sort of “catapult” that the multiplicative inverse acting on an infinitesimal will give.

    Applying to the cosmic space-time domain, we have a pattern of causal and temporal succession in finite stpes that are in principle countable.

    They come from the remote past of origins, and reach to the present. From, say, the singularity as 0 at 13.7 BYA or whatever, we have a finite span to the present, here n*. The issue is then beyond that horizon, where some claim completion of an actually infinite succession:

    . . . A, A ~ 1, A ~ 2, . . . r, . . . 2, 1, 0, 1*, 2*, . . . n*

    But the issue already identified is instantly applicable.

    The finite succession is complete-able, the transfinite is endless by definition, of different orders, aleph null being applied to counting numbers.

    And so, if we can already see that we have a problem bridging from A to r, there will be a problem bridging to A from the endless values beyond it.

    It seems to me that the best answer is, that there is some r a finite distance from 0, which is the terminus of the space time domain, i.e. it is inherently finite.

    As for handy-dandy cookie cutter definitions of sets and members, at this stage I am quite leery, so that is why I have reverted to speaking of ordered succession and counting numbers, with extensions to the zone of transfinite ordinals.

    Can you address my concerns?

    I extend appreciation in advance.

    KF

  76. KF: I’m only interested in part of what you are interested in, but I’m wondering, just to make it clear, if you agree with the two statements I wrote above:

    A: every element of the set of natural numbers N = {0, 1, 2, 3, … } is a finite number.

    B: the number of of numbers in the set N is a transfinite number aleph null.

    notice, I am NOT saying that aleph null is in the set somehow

    You do say that “notice, I am NOT saying that aleph null is in the set somehow,” which does bear on your agreement with either A or B. In fact, I think you are saying that you agree with B, and that aleph null is not a member of N.

    What about A: every element of the set of natural numbers N = {0, 1, 2, 3, … } is a finite number.

    Do you agree with that?

  77. Aleta, on the narrow point, I can see that every number k we can actually complete a count to and make a complete k-set {1, 2 . . . k} will be finite, and will of course have cardinality k. When the indefinite or transfinite traverse ellipsis comes in, that is where my concerns begin; with an endless count, there is not going to be any upper limit count number like that. So, I do have a concern about claim A, especially in connexion with claim B, which I would like to see resolved. Simply listing k = 1 + 1 + . . . 1 k times and there is a successor k + 1 will not help as the ellipsis here is not a transfinite traverse if the claim is k is finite. Beyond, I simply say, the question is interesting but not the end of the story. The onward issue is traversing the transfinite. KF

  78. You have a concern about A: what is it? If we are just looking at the natural numbers, I don’t see how “traversing the transfinite” is relevant.

    You write,

    Now, let us go to such a succession that goes on through ellipsis to the zone where we pick up w and its successors, w being omega the number that is the “first” ordinal of cardinality aleph null:

    {} –> 0, to {0} –> 1, to {0,1} –> 2, . . . k, . . .

    [a transfinite span, as above with counting numbers in succession]

    . . . w, w + 1, w + 2, . . .

    But in the set of natural numbers one would never “go through the ellipsis to the zone where we pick up w”. w is not a number in the set N, so there is no way we could reach it in order to go through it.

    So, to repeat, every member of the set N is a finite number. Why do you have a concern about that?

  79. Aleta, I am not at that time discussing a finite span but the transfinite one implicit in a claimed completed infinite past. On the set N, my concern is the claim every member is finite [thus subject to a finite count out to some k] when joined to the second claim that the cardinality of the whole is transfinite. KF

  80. On the set N, my concern is the claim every member is finite [thus subject to a finite count out to some k] when joined to the second claim that the cardinality of the whole is transfinite.

    But what is your concern? (I am not claiming a completed infinite past, so that is not the current topic.)

    A: every element of the set of natural numbers N = {0, 1, 2, 3, … } is a finite number.

    B: the number of of numbers in the set N is a transfinite number aleph null.

    You are using language that is different than mine, and I can’t tell whether we are saying the same thing.

    I’m saying “B: the number of of numbers in the set N is a transfinite number aleph null.”
    Aleph null is the name of the order of infinity possessed by the natural numbers, and is considered a “transfinite number”, which define different classes of numbers.

    But you write, “the cardinality of the whole is transfinite.” By the whole, I assume you mean the set of natural numbers. Yes, the cardinality of the set of natural numbers is the transfinite number aleph null. But if you are saying what I am saying, what is the concern. Are statements A and/or B above wrong, and if so why?

  81. Aleta,

    See what happens to your sets of numbers when a physicist gets hold of it.

    https://www.youtube.com/watch?v=w-I6XTVZXww

    First try to explain if and where there’s any error other than you might illogically disagree with the conclusion.

    -Q

  82. nm

  83. In his first step, with S = 1 – 1 + 1 -1 + …, he correctly points out that you get different partial sums depending on the number of terms: 1 for an odd number of terms and 0 for an even number.

    He then says we should just average the two, and call the sum 1/2. That is balderdash! S oscillates between two sums, so it does not converge, which is the only legitimate sense in which we can say an infinite series has a sum. There is no legitimate mathematical justification for saying the sum is 1/2. Since everything else builds from there, the rest is all wrong also.

    See here for a longer explanation: https://plus.maths.org/content/infinity-or-just-112, which interesting enough shows a quantum physics use of some very much more advanced mathematics that would imply that S = -1/12. Quantum physics struggles often with quantities which compute as infinite but in fact, obviously in the real world, aren’t: Feynman became famous for coming up with a way to get around this problem in certain kinds of situations.

    But I object to the guy using bad mathematics to make his point.

  84. Aleta, I have no objections to or concerns regarding B. My problem, as described and explained, is how A can be compatible with B, given that we are in fact describing the counting numbers and how we get to the cardinality of “counting sets” as I spoke of for convenience. Where it looks to me like the claim that any counting number k is k = 1 + 1 + . . . + 1 k times over, and may be exceeded by k + 1 (showing it to be finite), raises issues of ordinals and of the traversing of a transfinite span when one moves off on the open ellipsis. Decreeing and declaring that oh, transfinite ordinals are not naturals, does not help my concern a lot, when at the same time, it is held that the naturals are endless and can be so arranged that proper subsets are in 1:1 mutually exhausting correspondence with the whole — the very definition of being transfinite in cardinality. And, I am avoiding the standardised terminology but reverting to first steps as it seems there is a problem of how the standard terms will be understood/defined. KF

    PS: I have had occasion to complain of poof-magic mathematical hand waving by physicists on occasion; going all the way back to undergrad years.

  85. Glad you watched the video, Aleta,

    The odd thing is, that this mathematics is used in quantum mechanics as he pointed out in the textbook!

    Nevertheless, the clip also demonstrates the futility of using mathematical analogies to make an argument. Analogies are great for explanation, but that’s where it ends.

    -Q

  86. kairosfocus,

    Here’s one that I figured out in Junior High (as I’m sure lots of other kids did too).

    Let n = 0.999… (repeating)
    10n = 9.999…
    10n-n = 9.999… – 0.999…
    9n = 9
    n = 1

    Therefore 0.999… = 1.000….

    🙂

    -Q

  87. Q, thanks, indeed the decimal, place value system has two formally equivalent expressions for a whole number, the [n-1] .999 . . . and the [n].000 . . . KF

  88. Aleta,

    I have had a thought: if the point is, that there are unlimited numbers that can be set in finite counting sets, I can see that . . . just keep going in steps. However, that does not answer to another linked issue, the import of the ellipsis that the ordered sequence is unlimited in character.

    For that, setting the ordered numbers in line and applying a definition of a “border” as: if it is finite it is natural, and if it is beyond a transfinite span from 0, it is not a natural though it may be an ordinal starting from w, does not resolve the onward point on descent from the infinite past as claimed.

    Is that a step of progress?

    KF

    PS: In effect I read that as, as one moves out in steps from 0, things get into a vast impassable wilderness but the sequence keeps going, and eventually it arrives at w and so forth onwards way out there in numbers Sahara territory. Of course the y = 1/x wormhole near 0 allows one to catapult to that zone in one astonishing step. Once you are at the w zone, the same impassable Sahara faces s/he who would descend in steps to the zone near 0, so again, the catapult must be used. Now, somebody needs to go write a new Flatland.

  89. KF,

    Your post #90 echoes this statement you made earlier:

    The room inspection tour must start in the transfinite zone and arrive at the finite one in order to inspect all rooms.

    In the Hilbert Hotel model, the tour I described has no starting point. The hotel manager has been inspecting the rooms throughout the assumed infinite past. Furthermore, he has never been more than finitely many steps from the front desk.

  90. Hi Querius. You write, “Nevertheless, the clip also demonstrates the futility of using mathematical analogies to make an argument.”

    I agree with you about analogies in general – they are useful in illuminating ideas, sometimes, but don’t themselves ever prove anything.

    However, I don’ think the clip is using mathematical analogies – it is just using mathematics that is wrong.

    I

  91. DS, pardon but I said nothing of a starting point, I spoke to a starting zone. One that is transfinitely remote from the finite neighbourhood of 0. If the inspection arises at the transfinitely remote zone of the hotel, it has to traverse a transfinite span in single, finite steps if it is to reach a finite neighbourhood of 0, and then eventually rooms . . . -2, -1, 0. I chose a point in that zone, A, to begin a down-count of steps as the tour must pass each successive room. As he passes A, I start to count, to show why something based on an inherently finite stepwise incremental and cumulative process will not be able to traverse the intervening span onwards from A to reach a finite neighbourhood of 0. And nope, substituting a finitely remote zone relative to 0 does not answer the point. If you are inspecting an actual infinity of rooms in succession, you must start at the transfinitely remote zone. Such a tour will not complete as a stepwise process and will not ever reach a zone finitely remote from 0 as at every successive room past A the manager will only be a finite distance past A, never mind how that increments on and on. The implications for a claimed actually infinite in the past space-time world that proceeds in cumulative, finite causal steps, will be plain. KF

  92. Aleta, analogies are not proofs but are often highly instructive and inductively strong. By analogy, if I see you are evidently a mammal, much can safely be inferred by analogy with type-cases, never mind cases such as whales or platypuses. KF

  93. kf, you write, in three different posts, the following:

    In effect I read that as, as one moves out in steps from 0, things get into a vast impassable wilderness but the sequence keeps going, and eventually it arrives at w and so forth onwards way out there in numbers Sahara territory.

    raises issues of ordinals and of the traversing of a transfinite span when one moves off on the open ellipsis

    DS, pardon but I said nothing of a starting point, I spoke to a starting zone. One that is transfinitely remote from the finite neighbourhood of 0. If the inspection arises at the transfinitely remote zone of the hotel,

    All of these seem to say that if you examine the natural numbers N = {1, 2, 3 …}, at some point you get “past the ellipsis” into a “transfinite zone” in which you get to the transfinite numbers and “eventually arrive at w and so forth onwards way out there in numbers”

    All of these things are not true. If you limit yourself to the natural numbers, you never get “past the ellispsis.” The transfinite numbers are a different type of number, but they are not part of the natural numbers.

    I don’t believe there is anything in standard mathematics about the natural numbers that corresponds to what you are calling a transfinite zone.

    So to say that ” as one moves out in steps from 0, things get into a vast impassable wilderness but the sequence keeps going, and eventually it arrives at w” is wrong. If you move in steps out from 0 you just keep getting bigger and bigger natural numbers – you never get to some other “zone” and you never get to w.

  94. KF,

    A “zone” transfinitely remote from the finite neighborhood of 0? You will have to give a proper definition of this.

    Every single room in the HH I am describing is finitely remote from the front desk, so you won’t be able to do the above.

    Let me also address your use of ω.

    My HH has room numbers consisting of the opposites of the natural numbers only.

    In effect, you are working with an extended HH, say HH(ω) which has room numbers also including (some? all?) infinite ordinal room numbers (with a minus sign in front, if you will). That’s fine, but that’s not the HH I am describing a tour of.

    ω is not a successor ordinal. I cannot use my procedure of “backing up one room” from room -ω to get to an adjacent room closer to the front desk. Hence my procedure does not work to define a tour of HH(ω). But of course I never claimed I could.

    My tour traverses HH only.

  95. Aleta, I have not intended any surpassing of an ellipsis, save where that is specified as the continuation reaches a specified level. An open ended ellipsis implies unlimited extension. I do envision that there may be a transfinite span in a case with a two-sided ellipsis, as in say: 0, 1, 2, . . . w, w +1, w + 2 . . . for which the first is two sided and the second open on its RHS. This would be as opposed to a finite span 0, 1, 2 . . . k, k + 1 . . . where the first ellipsis would be finite in its span. Where also, w, for convenience, stands for the transfinite ordinal omega. And the issues at stake pivot on that question of traversing a span that is transfinite. KF

  96. DS, if every single room is finitely remote the total of rooms should be finite, finiteness at each step implies a thus far finite neighbourhood of 0. If the total of rooms never completes as a finite count and is actually infinite, it seems to me that it must therefore include a zone that is transfinitely remote from the value 0. That is why I look at an inspection that begins at the remote zone and needs to traverse the span to 0 in steps. KF

    PS: making a distinction in successive numbers of unlimited extent that “naturals” are finite and w etc as transfinite are not “naturals” does not to my mind eliminate the ordinal number pattern of unlimited succession. I am specifically interested in spans that go all the way to a transfinite zone and in what happens if one tries to ascend/descend in steps.

  97. kf, so you are saying that you can surpass the ellipsis “where that is specified as the continuation reaches a specified level.”

    What does that second phrase mean? What “specified level” do you reach that allows you to surpass the ellipsis? I know of no “levels” in the natural numbers, and no place in the continuation of going to the next natural number that the nature of the continuation changes.

    So can you explain more how the process of going to the next natural number ever reaches a “specified level” where we “surpass the ellipsis.”

  98. KF,

    It seems to me that you want to be a finitist, yet you accept the existence of objects such as π, ω and perhaps even noncomputable numbers.

    If the total number of rooms in the HH is finite, what is ω?

    Edit:

    PS: making a distinction in successive numbers of unlimited extent that “naturals” are finite and w etc as transfinite are not “naturals” does not to my mind eliminate the ordinal number pattern of unlimited succession. I am specifically interested in spans that go all the way to a transfinite zone and in what happens if one tries to ascend/descend in steps.

    That’s great, but I’m not that ambitious, for the purposes of this discussion. This is why you’re looking at a different problem from mine. You want to investigate traversals of HH(ω) while I am content to stick with HH.

  99. Hi kf. You added the following to your previous post:

    I do envision that there may be a transfinite span in a case with a two-sided ellipsis, as in say: 0, 1, 2, . . . w, w +1, w + 2 . . . for which the first is two sided and the second open on its RHS. This would be as opposed to a finite span 0, 1, 2 . . . k, k + 1 . . . where the first ellipsis would be finite in its span.

    What you envision is NOT the natural numbers. You are envisoning a sequence which includes both natural numbers and transfinite numbers, but they are two different types of things, and I don’t believe they can be put in sequence as you are doing.

    Also, you say “This would be as opposed to a finite span 0, 1, 2 . . . k, k + 1 . . . where the first ellipsis would be finite in its span.”

    If you are meaning to refer to the natural numbers here, that would NOT be a finite span. The ellipsis refers to the never ending process of going to the next natural number, which is an infinite span, not a finite one.

  100. And kf, you write, “I am specifically interested in spans that go all the way to a transfinite zone and in what happens if one tries to ascend/descend in steps.”

    But there is no “transfinite zone” in the natural numbers. You have created a sequence that includes the natural numbers and then the transfinite numbers as if the second somehow followed the first in an ordered set.

    But I don’t believe such a sequence is mathematically meaningful. Do you have a reference where this sequence is discussed in any mathematical literature?

  101. Correction to kf: I misread when you wrote, “Also, you say “This would be as opposed to a finite span 0, 1, 2 . . . k, k + 1 . . . where the first ellipsis would be finite in its span.”

    Yes, the first ellipsis would be finite.

    I was thinking of your other sequence “0, 1, 2, . . . w, w +1, w + 2 . . .”, which is a sequence that I don’t believe exists, and is mathematically not meaningful.

  102. Aleta,

    The ordinals don’t make up a sequence that is indexed by the natural numbers as sequences most often are, but the notation

    0, 1, 2, …, ω, ω + 1, ω + 2, …, ω + ω, …

    actually does make sense, in that the ordinals do form a totally ordered set. ω itself is greater than any of the finite ordinals, so it’s meaningful to arrange it after all the finite ordinals, after the first ellipsis.

    See this for more details.

  103. OK, that is interesting, although I don’t completely understand it.

    However, would you agree with my other comments that if you restrict yourself to the natural numbers you don’t “surpass the ellipsis” and get to the transfinite zone (which I presume means the w, w+1, w+2… sequence.

    That is, even if {0, 1, 2, …, ?, ? + 1, ? + 2, …, ? + ?, …} does list all the ordinals, the ?, ? + 1, ? + 2, …, ? + ?, … part is not part of the natural numbers?

  104. Aleta,

    Yes, I do agree with your other comments. I’ve been trying to say the same thing I think, specifically that ω is not the successor of any natural number, so there is no way to count up to or down from ω [Edit: from any natural number].

    I guess KF believes there is some murky quasi-infinite zone in the natural numbers, but as you have clearly pointed out, the order and magnitude properties of N are not that mysterious or exciting.

  105. DS, I suggest to you that there is a problem of perceptions. KF

  106. So many comments and we are back at: KF is right and DS is wrong. And if math agrees with DS, then math is wrong, too.

  107. KF: Well, can you clarify how our perceptions of your position are incorrect?

  108. HRUN, you were answered long since. KF

  109. DS, just for now, re 100, the hotel infinity cannot have a finite total of rooms. That is a clue as to some of the issues underlying the points at stake. KF

  110. kf, no one says that “hotel infinity”, aka, the natural numbers, have a finite number of rooms, aka numbers. They have an infinite number of elements, and that number is aleph null, the lowest order of infinity.

    But you keep talking about a “transfinite zone” that is “beyond the ellipsis”: in reference to the natural numbers, there is no way to get “beyond the ellipsis”, and there is no such thing as a “transfinite zone”. That is the issue at stake.

    Besides Dave’s posts, please see my 95, 99, 101. and 102.

  111. Aleta,

    it seems there are all sorts of gaps of communication at work; I can only pause a moment just now.

    There is a reason why I have spoken of an ordinal sequence of counting numbers, of counting sets and how such are finite but successively larger following the cardinality k where the kth set has 1 + 1 + . . . + 1 = k, with 1 additive step repeated k times and a finite span ellipsis in the notation.

    Obviously a finite step produced set cannot but be finite. But the interest is a transfinite set, for which the problem becomes that the ellipsis has to become of more than a finite span.

    And, ordinals come first, with w being the first transfinite and a successor to the finite ordinals.

    Let me (in part for HRUN’s benefit) clip Wolfram for the moment:

    http://mathworld.wolfram.com/OrdinalNumber.html

    In formal set theory, an ordinal number (sometimes simply called an “ordinal” for short) is one of the numbers in Georg Cantor’s extension of the whole numbers. An ordinal number is defined as the order type of a well ordered set (Dauben 1990, p. 199; Moore 1982, p. 52; Suppes 1972, p. 129). Finite ordinal numbers are commonly denoted using arabic numerals, while transfinite ordinals are denoted using lower case Greek letters . . . .

    The first transfinite ordinal, denoted omega [–> I have been using w], is the order type of the set of nonnegative integers (Dauben 1990, p. 152; Moore 1982, p. viii; Rubin 1967, pp. 86 and 177; Suppes 1972, p. 128). This is the “smallest” of Cantor’s transfinite numbers, defined to be the smallest ordinal number greater than the ordinal number of the whole numbers. Conway and Guy (1996) denote it with the notation omega={0,1,…|}.

    From the definition of ordinal comparison, it follows that the ordinal numbers are a well ordered set. In order of increasing size, the ordinal numbers are

    0, 1, 2, …, omega, omega+1, omega+2, …, omega+omega, omega+omega+1, ….

    The notation of ordinal numbers can be a bit counterintuitive, e.g., even though 1+omega=omega, omega+1>omega. [–> in the notation, commutativity is broken] The cardinal number of the set of countable ordinal numbers is denoted aleph_1 (aleph-1).

    If (A,LT =) is a well ordered set with ordinal number alpha, then the set of all ordinals LT alpha is order isomorphic to A. This provides the motivation to define an ordinal as the set of all ordinals less than itself . . . .

    There exist ordinal numbers which cannot be constructed from smaller ones by finite additions, multiplications, and exponentiations. These ordinals obey Cantor’s equation. The first such ordinal is epsilon_0 [–> I have used E_0] . . . .

    Ordinal addition, ordinal multiplication, and ordinal exponentiation can all be defined. Although these definitions also work perfectly well for order types, this does not seem to be commonly done. There are two methods commonly used to define operations on the ordinals: one is using sets, and the other is inductively.

    Immediately we see an explanation of many concepts and constructs and in particular the presence of an ellipsis of transfinite span:

    0, 1, 2, …, omega, omega+1, omega+2, …

    with the finite ordinals in the neighbourhood of 0 succeeded by an ellipsis of transfinite span and this by w, then w + 1 etc, with an onward ellipsis of likewise transfinite span.

    It is instantly clear that such a transfinite span cannot be traversed and spanned in a succession of finite steps, whether ascending or descending [and hence the use of the general terms level and zone].

    Earlier today, I spoke of naturals, k as finite counting sets, and how I could see an endless incremental succession of same. That endlessness of succession is where the issue of transfiniteness enters.

    Maybe, we can now revisit other points, clarifying along the way?

    Pardon, I have to run just now.

    Later.

    KF

  112. Aleta, I am responding to something projected unto me in comment 100. KF

  113. KF,

    All this leads me to ask, if I describe to you a set, how could you determine that it is in fact infinite?

    For example, the HH. How could you know it has infinitely many rooms? At any “stage”, your counting process has progressed only to a finite number.

    Consider the set of all numbers 2^k, where k is a natural number.

    Is this an infinite set? It’s in 1-1 correspondence with N of course, so I say yes.

    But I think you’re going to get hung up at the same place that caused you to conclude that the set of rooms in the HH at finite distance from the front desk is finite.

    I don’t think your views on counting allow infinite sets at all.

  114. kf, the sequence of ordinals mentioned in Wolfram, 0, 1, 2, …, omega, omega+1, omega+2, …, etc. is NOT the same as just addressing the set of natural numbers. w, w + 1, etc. are not in the set of natural numbers and are not somehow “beyond the ellipsis”. That sequence is not relevant to discussing taking unit steps on a number line, traversing the natural numbers

    You write,

    “Obviously a finite step produced set cannot but be finite. ….

    I spoke of naturals, k as finite counting sets, and how I could see an endless incremental succession of same. That endlessness of succession is where the issue of transfiniteness enters.

    Herein possibly lies the confusion.

    If we consider the set of numbers K = {1, 2, 3 , … k}, then indeed that set is finite.

    If we consider the set of numbers N = {1, 2, 3, …}, that set is infinite in size even though each member is a finite number. The only place transfiniteness show up in this discussion is that aleph null is the name given to the level of infinity possessed by the naturals, The sequence in Wolfram is NOT about further numbers in N that are in some further “transfinite zone beyond the ellipsis.” N = {1, 2 3, …} does NOT eventually include w.}

  115. DS, as I already noted above per standard results a set will be transfinite when it can be placed in 1:1 correspondence with a proper subset. KF

  116. Aleta, I have been deliberately staying away from debating the naturals and have addressed the ordinals starting from von Neumann’s construction on {} up and have above discussed ordinals across a transfinite span indicated by an ellipsis. With the Wolfram discussion and direct parallel to my earlier remarks in play in answer to doubts and dismissals that may be inspected above. At no point in our current discussion have I said w and on are natural numbers, but I have said they form an ordinal scheme extending from 0, 1, 2 etc on. Whether or not the natural numbers terminate before that level and whether or not all natural numbers are finite [but have an overall cardinality that is transfinite*], the transfinite ordinals w etc are a continuation from 0, 1, 2 . . . as counting numbers. That is all I need for the issue of needing to traverse the transfinite to become a serious question on issues regarding a claimed infinite past. KF

    *PS: The counting scheme k = 1 + 1 + . . . 1 k times raises interesting questions on what is in that ellipsis in all cases.

  117. KF,

    DS, as I already noted above per standard results a set will be transfinite when it can be placed in 1:1 correspondence with a proper subset. KF

    So the set of finite nonpositive integers is infinite because we have a 1-1 correspondence between it and one of its proper subsets. f(n) = n – 1 is one such correspondence.

    If you agree with that, then this also gives a 1-1 correspondence between the set of room numbers for rooms at finite distance from the front desk and one of its proper subsets.

    Therefore the set of rooms finitely distant from the front desk is infinite.

  118. kf says,

    Aleta, I have been deliberately staying away from debating the naturals.

    I’ve noticed that.

    However it is the natural numbers that are relevant to the hotel example, and it is the natural numbers that are relevant to the discrete steps of a “step by step causal succession” n time. The model here is the number line, with the natural numbers (or the integers if you wish) as the things we are talking about, and with the number line there is no “bridge to the transfinite” involved. The sequence of ordinals is not relevant to either the infinite hotel nor the idea of time passing in a step-by-step causal way.

    So you have been deliberately staying away from exactly the topic that is relevant.

  119. DS, the problem is not that the whole numbers form an endless continuation, it is that finitude at given k on this particular set will mean that the span to k is finite not transfinite. For the span to be transfinite it has to be just that, limitless. Also, I would think that a lagged count of the set will not be a proper subset. As the put in infinitely many fresh guests by putting current room n occupant to 2n and new ones to 2n – 1 shows, the whole matches the evens. And no finite span of rooms will be such that transfer of the full house guests from n to 2n will accommodate all existing guests. It has to be endless. KF

  120. KF,

    Also, I would think that a lagged count of the set will not be a proper subset.

    Eh? The function I defined gives a 1-1 correspondence between A = {…, -2, -1, 0} and B = {…, -3, -2, -1} = A – {0} (set difference).

    B is a proper subset of A. Therefore A is infinite.

    Note also that A is the set of room numbers for rooms finitely distant from the front desk (counting the front desk as a room).

  121. kj, Hilbert’s Hotel is about ” a hypothetical hotel with a countably infinite number of rooms” (wikipedia and Wolframs’), and “countably infinite means having the same cardinality as the natural numbers.

    Therefore, if you aren’t discussing natural numbers, you are not discussing Hilbert’s Hotel.

  122. Yes, kf, the natural numbers are endless. However, they don’t pass the ellipsis into the transfinite zone”, whatever that means.

    They are defined by the fact that for every natural number k, k + 1 is also a natural number. Each k is a finite number, and there are an infinite number of them. Those are really the only two points I’ve been trying to discuss with you.

  123. KF,

    I didn’t read this carefully:

    As the put in infinitely many fresh guests by putting current room n occupant to 2n and new ones to 2n – 1 shows, the whole matches the evens.

    Yes, this is another proof that the set of rooms finitely distant to the front desk is infinite.

  124. Aleta, for each finite k, the room count will be just that, finite, which will iterate for k + 1. Kindly now continue that essential finiteness of successive steps and show us how you obtain cumulative transfiniteness of the count [the type order, I believe is the technical term] without attaining to at least w in the ordinal succession, which is where cardinality aleph null kicks in. Where, w is the first transfinite ordinal. KF

  125. DS, kindly start with the full hotel. Infinitely many new guests pull up. Our manager shifts current guests in rooms n to 2n, and puts the new ones in 2n – 1; none of the old guests are roomless and the new ones are also now in rooms though the hotel was formerly full. Let this be seen to happen where the number of rooms is not transfinite — i.e. endless — attaining to at least w in the sequence as would be in effect painted on the doors. w is the first transfinite ordinal, of cardinality aleph null. Kindly explain. KF

  126. kf, Cardinality aleph null doesn’t “kick in”, and you don’t ever attain “w in the ordinal succession”.

    Aleph null is a name for the size of an infinite set equivalent to the natural numbers. It is not a place or quality that you aren’t at for a while and then you are. You, the person taking the steps, can never achieve transfiniteness, even though the set you are traversing is infinite. Each natural number has also an ordinal position equivalent to its cardinal value, but there is no place within the natural numbers where the ordinal value w is reached.

    This is a number line, kf. You can walk forever, and you’re still on the number line. That’s all. There is no “transfinite zone” on the number line.

  127. kf writes,

    Let this be seen to happen where the number of rooms is not transfinite — i.e. endless — attaining to at least w in the sequence as would be in effect painted on the doors. w is the first transfinite ordinal, of cardinality aleph null. Kindly explain. KF

    First a question, kf: are you using transfinite to just mean infinite in the common mathematical sense? Or does it have some more and/or other meaning to you?

    Second, are you saying that if the room numbers were painted on the doors, there would be a door someplace with w written on it?

    You write: “w is the first transfinite ordinal, of cardinality aleph null.” – true.

    However w is not a number, or a division point, or anything on the number line. There is no place in the infinite hotel that corresponds to w. w says something about the size of of the hotel, but it does not describe any place in the hotel.

  128. KF,

    I think you have just explained it. Nowhere in the re-rooming scheme did you mention any ω’s or otherwise transfinite numbers.

    The original occupants are moved to the even-numbered rooms, and the new guests are interleaved among them in the odd-numbered rooms. We’re talking about even and odd natural numbers, which are all finite.

    Did you think they would run out of room? That’s not a problem with an infinite hotel.

  129. Aleta (and DS),

    my first point is that once we have endlessness, there is an emergent state — the omega point if you will. But that is precisely that, emergent and organically tied to what has gone on to reach that point.

    The nature of endlessness is that it can never be a completion and termination of a finite step by step process.

    Which, is directly tied to the focal context for all of this: the naturalistic claim of an actualised real world infinite temporal past as an alternative to the equally naturalistic claim of a world from utter non-being, a genuine nothing.

    We can create a real world step by step process that continues and for all we see is POTENTIALLY endless but at every stage an actual finite step based process will be finite and have a k steps so far character.

    By contrast, we may provide a conceptual pattern of ordinal numbers that indicates and provides numerals for the transfinite, ordinal and cardinal.

    And, BTW, transfinite is Cantor’s term to confine discussion to the subject in hand as opposed to all that may be inadvertently dragged in when we use the term infinity.

    I go to the real world claim.

    An infinite actual past bridged to the present implies traversing a transfinite span in cumulative discrete finite causal steps that may be enumerated. That is, inter alia the claim is subject to the scrutiny of the logical study of structure and quantity, aka mathematics.

    I have repeatedly used a mathematical catapult to get us to the transfinite domain, based on the continuum in the closed interval [0, 1].

    Similar to how Robinson et al defined hyper reals including hyper integers, let us go to some m –> 0 but is not quite 0. An infinitesimal of a sort. Let us extract its multiplicative inverse, further stipulating that it will give us a whole number type result, fractional part uniformly zero. 1/m = A, where A = w + g, a successor to omega where g is some large but finite value.

    So, we can represent, with k a finite value:

    {} –> 0, to {0} –> 1, to {0,1} –> 2, . . . k, . . .

    [a transfinite span, as above with counting numbers in succession]

    . . . w, w + 1, w + 2, . . . w + g [g a large finite value], …, w·2, w·2 + 1, …, w^2, …, w^3, …, w^w, …, w^[w^w], …, E_0, ….

    It matters not that many successors to A exist, our interest is its predecessors as in A less one, [w = (g – 1)] etc. Then we may symbolise that as A~1, A~2, etc.

    Where, the cardinality of A is the same as w, aleph null. The first degree of endlessness.

    From that, this is what I have done to draw out the issue of having to traverse a transfinite span in a stepwise descent to a finitely remote neighbourhood of 0 followed by descent to 0 and ascent from 0 with intent to assign 0 as the singularity and some n* as now. I clip 77 above from Feb 2:

    it seems, we may reasonably list in reverse form:

    . . . A, A less 1 [i.e. W + (g-1)],. . . w, . . . 2, 1, 0

    Thus — without assuming that there is anything but an indefinitely large span beyond A to the left — we can look at down-counting from A, symbolising [w + (g – 1) as A ~1, etc:

    A, A ~ 1, A ~ 2, . . . w + 1, w, . . . r, . . . 2, 1, 0

    0#, 1#, 2# . . .

    Now, g is finite but very large, this allows us to establish an ordering down to w.

    My key concern is that the ellipsis beyond w is endless, so the stepwise down count sequence that begins at A will go on forever without bridging it to r, a finite neighbourhood of 0.

    Indeed, it seems that once the secondary count started at A hits w at g steps later, and goes on, we are in the position of trying to count across an endless span.

    Thus, a stepwise count process will not reach r, much less the interval [0,1].

    To me, this seems to give some substance to the remark that an inherently finite stepwise counting process will not bridge a transfinite span, will not traverse a transfinite range.

    This is as distinct from that we may set in order a succession that as a set will define a transfinite span of ordered succession. Countable in principle, inexhaustible in practice. The gap between the potentially infinite process and the actually completed infinite process.

    To span the transfinite, it seems to me we need the sort of “catapult” that the multiplicative inverse acting on an infinitesimal will give.

    Applying to the cosmic space-time domain, we have a pattern of causal and temporal succession in finite stpes that are in principle countable.

    They come from the remote past of origins, and reach to the present. From, say, the singularity as 0 at 13.7 BYA or whatever, we have a finite span to the present, here n*. The issue is then beyond that horizon, where some claim completion of an actually infinite succession:

    . . . A, A ~ 1, A ~ 2, . . . r, . . . 2, 1, 0, 1*, 2*, . . . n*

    But the issue already identified is instantly applicable.

    The finite succession is complete-able, the transfinite is endless by definition, of different orders, aleph null being applied to counting numbers.

    And so, if we can already see that we have a problem bridging from A to r, there will be a problem bridging to A from the endless values beyond it.

    It seems to me that the best answer is, that there is some r a finite distance from 0, which is the terminus of the space time domain, i.e. it is inherently finite.

    As for the Hotel Infinity, the procedure for adding infinitely many new guests, moving present ones to the even rooms 2n, n being present room, and adding the infinity of new guests to the odd ones 2n – 1; that can only work if the number of rooms is transfinite. The hotel as a whole expresses w.

    That is, it seems that the relevant point is that w is in effect the numerical designation of the first emergence of endlessness. Thus, it can be viewed as an emergent property of endlessness.

    So, yes, any counting number k we can write down will be finite as product of 1 + 1 + . . . 1 k times over, but once we have an endless succession, it seems to me that an emergent effect is that the endless succession as a whole manifests w, the first transfinite ordinal.

    From that we may proceed in the transfinite zone.

    And the endlessness involved means we cannot actually complete the ascent to w in successive real world steps. Nor can we carry out a descent from there in that way.

    The transfinite span ellipsis is highly significant.

    On this, I think I can accept that any whole number k we can write out or count up to in steps will actually be finite, but that endless succession — which we cannot actualise in fact but only indicate in principle — creates an emergent property, transfinite numbers.

    That is, as we lay out the number line and continue the ordinal sequence with an ellipsis that goes on to w, there is an implied transfinite span. So also {0, 1, 2, . . . } implies a transfinite span and the whole as a set expresses w.

    And once endless succession lurks in an ellipsis, we will need to face the issue of traversal of a transfinite span and the futility of claimed or intended actual traversal in successive finite steps.

    KF

  130. KF,

    Similar to how Robinson et al defined hyper reals including hyper integers, let us go to some m –> 0 but is not quite 0. An infinitesimal of a sort. Let us extract its multiplicative inverse, further stipulating that it will give us a whole number type result, fractional part uniformly zero. 1/m = A, where A = w + g, a successor to omega where g is some large but finite value.

    No matter how small your positive real number m is, then 1/m will not be a successor to ω. It’s an impossibility. You will simply get a real number. The multiplicative inverse of any nonzero real m is also real, and not an infinite ordinal. Refer here to the field axioms.

    This has been pointed out many, many times, so I don’t understand why you keep posting this “construction”.

    And the endlessness involved means we cannot actually complete the ascent to w in successive real world steps. Nor can we carry out a descent from there in that way.

    Note that there is no “ascent to ω” in any of these HH puzzles. The guest originally in room 10^150 gets moved to room 2*10^150; one of the new guests is placed in room 2*10^150 – 1. All the room transfers go like this, with each hotel occupant ending up in a room with a finite number.

    On this, I think I can accept that any whole number k we can write out or count up to in steps will actually be finite, but that endless succession — which we cannot actualise in fact but only indicate in principle — creates an emergent property, transfinite numbers.

    I would more or less agree with this.

    That is, as we lay out the number line and continue the ordinal sequence with an ellipsis that goes on to w, there is an implied transfinite span. So also {0, 1, 2, . . . } implies a transfinite span and the whole as a set expresses w.

    Well, nothing is “going on to ω here”. The statement that the entire set expresses ω sounds reasonable.

    And once endless succession lurks in an ellipsis, we will need to face the issue of traversal of a transfinite span and the futility of claimed or intended actual traversal in successive finite steps.

    Except perhaps in the case of an eternal process extending through an infinite past, such as a beginningless ticking clock or tour by the manager of the HH.

  131. DS, as has been repeatedly put forward across this discussion, m is an infinitesimal of a mild enough degree to land us at w + g through its multiplicative inverse. The point is, [0,1] is a closed, continuous interval so there will be numbers closer to 0 than any epsilon-neighbourhood you can construct, i.e. infinitesimals are there on the continuuum. Such are real enough to found a whole alternative approach to calculus — nonstandard analysis. KF

    PS: step by step traversal of the transfinite is a supertask, and as such the reasonable conclusion is the proposal of an infinite actual past is futile. Concluding an infinite count down to the singularity then upwards a finite count to the present will fail due to the problem of the transfinite spanning ellipsis. Beginning is not the pivotal issue, completion is. Where the point of A, A~1 etc above is to show that continuation step by step to try to access 0 is futile.

  132. KF,

    DS, as has been repeatedly put forward across this discussion, m is an infinitesimal of a mild enough degree to land us at w + g through its multiplicative inverse. The point is, [0,1] is a closed, continuous interval so there will be numbers closer to 0 than any epsilon-neighbourhood you can construct, i.e. infinitesimals are there on the continuuum.

    This is false. The real number field is Archimedean, which means that for any positive x in R, there exists a natural number n such that 1/n is less than x.

    In other words, x does not lie in the neighborhood (-1/n, 1/n) of zero.

  133. DS, go to the nonstandard analysis. Infinitesimals are valid near-0 members of the continuous closed interval [0,1] for purposes of logical analysis of structures and quantities. KF

    PS: If you find a catapult approach unacceptable, try simply A = w + g, g a finite. A is of cardinality aleph null.

  134. KF,

    Then as Aleta pointed out, you’re no longer talking about traversing the HH.

    The infinitesimals and nonreal hyperintegers have no bearing on that problem.

  135. Hi kf.

    You write,

    As for the Hotel Infinity, the procedure for adding infinitely many new guests, moving present ones to the even rooms 2n, n being present room, and adding the infinity of new guests to the odd ones 2n – 1; that can only work if the number of rooms is transfinite. The hotel as a whole expresses w.

    You say above “only work[s] if the number of rooms is transfinite.” But the number of rooms is infinite, with cardinality aleph null – that is the premise of the whole analogy.

    The hotel analogy is just about the nature of the natural numbers – no “transfinite zones” needed.

    That is, it seems that the relevant point is that w is in effect the numerical designation of the first emergence of endlessness. Thus, it can be viewed as an emergent property of endlessness.

    What does the “first emergence of endlessness mean?” I am walking down the number line. At every number I can always take the next step: at k there is always k + 1. Given that there is always a next step, the procession is always endless, starting at my very first step.

    And I could go … (endlessly, it seems).

    But I won’t.

    kf, are you aware that your thoughts on this matter – the nature of infinity in respect to the natural numbers, is not standard mathematics and is not shared by other mathematicians. You have idiosyncratic and confused notions, and further discussion isn’t going to change anything.

    And you don’t need your notions to make the point you want to make. You claim that we can’t get to infinity, or get here from negative infinity, in a finite number of steps. That is true – that is what infinity, the ellipsis, means. There is no need to talk about “bypassing the ellipsis to get to a transfinite zone” to make that point.

    So, I am going to stop my role in this discussion at k, recognizing that if I go to k +1 I’ll just have to go on to k + 2, and so on (…) without any further progress or enlightenment being made.

    For what it’s worth, I’ve learned some things (or at least deepened past understandings). I remember first being introduced to these ideas about infinity back in college calculus classes, and of course dealt with the nature of natural and real numbers while teaching calculus for many years. Reading the articles at Wikipedia and Wolframs, as well as the posts by others here and elsewhere, has been interesting.

    So I’ve both enjoyed the discussion for it’s own sake, as well as for the usual challenge of trying to have a focused conversation with kf.

    But it’s time for me to move on for now.

  136. KF,

    One technical point: The infinite hyperintegers are different from the infinite ordinals/cardinals from set theory, so there is no hyperreal infinitesimal m such that 1/m is an infinite ordinal number.

  137. As the HH hotel manager goes from room to room, does he leave the light on?

  138. Aleta & DS,

    First, I have consistently underscored that Hilbert’s Grand Hotel Infinity [HGHI] is exactly that, infinite. I think in that context, there is a clear tendency to so focus on the finite step by step sequence of natural numbers in looking at it that sight of that premise or context may be lost.

    So, it needs to be emphasised, it is HGHI henceforth.

    Second, that means, the number of rooms is endless and that it manifests, as a countable, ordered setting, the phenomena of w and of aleph null. While no particular room may be so labelled, the pattern of endless rooms makes such to be present.

    This further means that when one goes from 0 to 1 to k to k + 1 etc, there will be an ever extending onward path, yes.

    Now also, what is further very relevant is the issue of a transfinite down count.

    That is, there is a claim on the table of an infinite past that has somehow arrived at today.

    That is not going away as an issue, however personalities or concerns as expressed may be perceived or however arguments may be counter-pointed or may be dismissed.

    The focal issue is, that down count as the manager of HGHI supposedly has been endlessly inspecting rooms at the rate of one per second forever past and is claimed to now be say at room -r, a finitely remote neighbourhood of room 1. Say:

    . . . M –> -r . . . -2, -1, 0.

    Room 0 is the reception area etc.

    That down count must pass from a transfinitely remote zone to the current situation — if the infinity of new guests can be brought in by switching current ones to even rooms and putting the new ones in odd rooms, the HGHI must be such that there are endless rooms. If the countdown is real (or at least model-real), it must be complete to the current situation. That puts focus on the left hand ellipsis that leads the reversed sequence.

    So the reasonable question arises, how can the proposed state be? How can one traverse a transfinite span in steps to reach a finite neighbourhood of 0? Or, can one?

    Linked, what does an infinite actual past of the observed physical cosmos and/or its causal antecedents considered as strictly physical/material entities — as is being claimed by some — mean?

    Is it coherent, what would be its evidence, what would we find that can test such a hypothesis, etc?

    The first challenge is that this will confront the mathematics of the transfinite in order to be coherent, thus in the end, the ordinals. It is highly significant, in that context, to see the resistance and even dismissiveness in the thread above to the ordering pattern put up from

    0, 1, 2, . . . k, k +1 . . . w, w + 1, . . .

    at least, until there was citation from Wolfram, on which there has been mostly a silence rather than a response.

    If that transfinite ordering of numbers is valid (formerly doubted and dismissed above, now implicitly conceded), of course its opposite or reverse will also be valid:

    . . . w + 1, w, . . . k + 1, k, . . . 2, 1, 0

    (All that has here changed is the direction of the ordering. The fist two ellipses are transfinite in span indicated and the last one is finite.)

    Now, there was a second issue, in effect what does the succession mean, especially given the joint claims that all natural numbers in the set {0, 1, 2, . . . } are finite yet the set as a whole is of transfinite cardinality aleph null, which on the ordinal side is that of was first transfinite ordinal. In that context, there is a significant issue of identifying finite k as 1 + 1 + 1 + . . . + 1 k times, which implies that the ellipsis is addressing a finite span.

    This by no means implies that it is good enough to in effect say as every k is therefore finite there is nothing more to discuss; all “natural numbers” are finite even though the set as a whole is not; and just go away with — pardon, I here suggest a rhetorical effect not an inferred, intended persuasive effect — your confused notions.

    (I ask, beyond a loaded ellipsis, how is the transfinite character manifest in the ordered sequence 0, 1, 2 etc as it mounts up to be the set of naturals? How does the weight of w affect the whole? How do we express that beyond an ellipsis that invites, oh, it just goes on? How does the difference between finite and infinite manifest? Is saying all naturals are finite anywhere near an adequate statement in context? Is oh the set as a whole is transfinite notwithstanding capture the “what more” that needs to be said and appreciated?)

    I think, at minimum, there is something that needs to be made clear and shown coherent.

    Here, the issue is, we are looking at something that is a suggested model of the succession of causal chains that are claimed to be a consequence to date of a transfinite process. And the sort of ordinal line model is patently relevant, indeed, we can see that by simply suggesting that successive causal entities in the chain C be subscripted, C_k, etc. That is they are to be identified as in a succession that is held to go:

    . . . C_k +1, C_k, . . . C_2, C_1, C_0, C_1*, C_2*, . . . C_n*

    Where, C_n* denotes the current situation.

    But the natural question is, what does the leftmost ellipsis mean relative to a line that we could “expect” to have been something like:

    . . . C_w+1, C_w . . .

    [transfinite onward span, i.e. endlessness]

    . . . C_k +1, C_k, . . . C_2, C_1, C_0, C_1*, C_2*, . . . C_n*

    But in the wider pattern, we have two sets of ellipses that are addressing transfinite spans. The last two are relatively unproblematic, they are finite.

    From the discussions and context, it seems that the best so far conceptual understanding is that as the sequence of counting numbers increases without limit — and that is a qualitative change from oh individual numbers we represent as a finite chain of increments from 0, k, are finite — that leads to aleph null as in effect the emergent property, the first order of magnitude of limitlessness.

    So, once endlessness is involved, w and friends are present, even if there is no explicit symbolisation.

    I therefore say, we must look at the sequence of the transfinite ordinals, w, w + 1 as being involved with endlessness of counting numbers, etc.

    Where, Wolfram sums up:

    The first transfinite ordinal, denoted omega [–> I have used w for convenience], is the order type [–> in effect the “length” from 0 of the set of counting numbers so far] of the set of nonnegative integers (Dauben 1990, p. 152; Moore 1982, p. viii; Rubin 1967, pp. 86 and 177; Suppes 1972, p. 128). This is the “smallest” of Cantor’s transfinite numbers, defined to be the smallest ordinal number greater than the ordinal number of the whole numbers. Conway and Guy (1996) denote it with the notation omega={0,1,…|}.

    That is the succession of sets from {} –> 0, {0}–> 1, {0,1} –> 2 that incrementally increases without limit at that point is at the successor value w, which some folks represent {0,1,…|}. From w we may proceed at the transfinite succession to E_0 and so forth.

    Now, we can see that endlessness is the pivotal contrast to finitude. Often, expressed by a transfinite span ellipsis.

    I think we need to take endlessness quite seriously and at the force of its full meaning: having no completion, no terminus. With w present as a direct systemic result.

    So, instantly, there is an issue that if the chain of order does go to the transfinite in ascending mode, it must also be of the same order in descending mode.

    Therefore, it seems that a fair conclusion is that a transfinite span cannot be spanned or traversed in step by step finite, finitely incremented stages.

    So, it is not good enough to blandly assert that manager M of HGHI has been ticking off rooms like a clock forever, or that there has been a transfinitely successive causal chain of the physical cosmos that leads to now. At least, without serious explanation of the bridging of the transfinite, the completion of an endless succession.

    That has all the marks of a contradictory claim, especially for the physical world.

    Endlessness completed is not endlessness.

    It is ended.

    Spitzer is right to raise a red flag.

    But, this also points to the issue, how would that appear in our symbolism, and the answer is, there is a transfinite ellipsis that cannot be bridged in stepwise increments of unit size. To my mind this also means that claimed proofs of the naturals all being spanned by finite unit increments, and exceeded by the next in succession are running into the issue of the transfinite span. The potentially endless but not actualisable span.

    I think therefore that it is not credibly feasible for M to have spanned the transfinite in steps and thus have reached k — a finite range from 0. Likewise, it is not credible for the observed cosmos to have had an actual past transfinite succession of finite causal entities and causal transitions to the next stage, down to the present.

    Instead, it is credible that at some finitely remote k, there was an initiatory terminus. Matters not if that is 10^17 or 10^25 or 10^1700 s past, or whatever. Credibly, k is there.

    Of course, on the current understanding of the observed cosmos, there is such a beginning point,the singularity.

    There is speculation on a wider multiverse and on an endless past, though there is not a solid answer to observability or to the issue of ever rising entropy, which leads to the increasing non-availability of energy concentrations that can drive causal processes relevant to the world we observe around us. This is the concept sometimes termed heat death — at least, it is comparable to it.

    Now, too, I find it interesting to see the way there is an attempt to rule a datum line against the use of the catapulting effect of the hyperbolic function y = 1/x near 0, to span to the transfinite, by direct comparison to the approach taken in nonstandard analysis with the hyper reals and hyper integers as one manifestation.

    But in a world of calculus praxis, that is a very reasonable consideration. [0,1] is a closed interval, and is continuous. So it is reasonable to look at infinitesimals, m of mild enough character that they would catapult us to a range that is transfinite. (I take it from L’Hospital’s rule that we accept that infinitesimals can be of diverse order of relative magnitudes.)

    Yes, there is a debate on the relationship of the continuum to aleph null and successors, etc. The line of successive counting numbers invites filling in by the steps to the reals via rationals and it is notorious that it points onwards endlessly.

    So, how do we have coherence of the whole?

    I have suggested the simple step, 1/m = A = w + g, g a finite increment onwards from w.

    If we deal with truth, the issue of the general coherence of truth is there.

    And, as there has been so much of foreclosing discussion by appealing to the finitude of “all” naturals, I have felt it wise to back away and go to the first principles of counting numbers and constructing ordinal sets instead. That is not evasion of focal issues, but instead trying to address them in a way that does not needlessly run into perceptual barriers.

    I assume there is not incoherence in the logic of quantity and structure.

    But it is apparent that our understanding, individual and collective, is apparently fragmentary and thus incomplete.

    Notoriously, what is c and how it relates to aleph null, aleph 1 etc have been found to be undecidable.

    But c is the very stuff of the calculus, and it is tied directly to reals which in turn are traceable to counting numbers and fractions with power series used to give expression to fractional parts that my converge to values not finitely expressible in place value notation, e.g. pi or e etc.

    Where, place value notation is disguised power series notation. Which in turn clearly makes strong use of ordinals in expressing the sequence of digits, bases and powers.

    So, all of this connects to very commonplace things, and to all sorts of very real world issues.

    As I conclude for the moment, I draw attention again to the point that endlessness of succession of ordinals from 0 is important, and that once that endlessness is on the table — typically by implicitly transfinite ellipsis, once it first appears, we are dealing with the transfinite, however implicitly.

    That naturals — counting numbers form 0 to 1 to 2 etc — range on endlessly entails endlessness and so also the implications of the transfinite. Thence we face the impassable zone (at least for stepwise processes) illustrated by the transfinite ellipsis.

    And so, causal successive processes will be confronted with the logic of endlessness, whether from a start point ranging onwards, or whether we look at a claimed endless preceding succession that finds terminus in the now.

    So, whatever back forth exchanges, whatever conceptual struggles, whatever oh no it’s not that, then could it be like this then, etc, are important as we look onwards.

    This is not a cut and dried, settled commonplace only an idiot does not understand the way we do matter, for sure.

    KF

  139. KF,

    That’s … a lot of words. Let me try and address some of the substance.

    The first challenge is that this will confront the mathematics of the transfinite in order to be coherent, thus in the end, the ordinals. It is highly significant, in that context, to see the resistance and even dismissiveness in the thread above to the ordering pattern put up from

    0, 1, 2, . . . k, k +1 . . . ω, ω + 1, . . .

    at least, until there was citation from Wolfram, on which there has been mostly a silence rather than a response.

    Well, the ω’s are never traversed by the hotel manager, so I don’t know what we are required to say about the entire sequence. I acknowledge that it exists.

    The relevant sequence here is 0, 1, 2, …, or counting down, …, 2, 1, 0. These (or their opposites) are the rooms visited by the manager. I have to tell you when he was in room -n for each natural number n, which I have done. What more do you need?

    This by no means implies that it is good enough to in effect say as every k is therefore finite there is nothing more to discuss; all “natural numbers” are finite even though the set as a whole is not; and just go away with — pardon, I here suggest a rhetorical effect not an inferred, intended persuasive effect — your confused notions.

    Uh-oh. Scare quotes around the term natural numbers? Our confused notions??

    Aleta and I are using the standard definition of N.

    You’ve been given at least two proofs showing that every element of N is finite. One written by arguably the greatest living mathematician. IIRC, it was about 3 sentences long. Both were utterly elementary and no errors were pointed out by you.

    So yes, I take the finitude of every natural number to be settled.

    And I thought I just saw you acknowledge that this could be true somewhere in this post, but I can’t find it now.

    So that we both understand each other, is it true that every natural number is finite? Yes/No?

    I therefore say, we must look at the sequence of the transfinite ordinals, w, w + 1 as being involved with endlessness of counting numbers, etc.

    “involved with” is not a very specific claim, to say the least. The rest of your statements are so vague, there’s really nothing I can say. It’s like wrestling with fog.

    So, I suggest that as you have now sketched out your ideas, write this up more carefully and consult a trained mathematician and/or submit it to a journal!

  140. DS, just to begin your manager would have to traverse a transfinite interval to reach a finite neighbourhood of 0. That is the core issue of the ellipsis which you have consistently not addressed, and this requires completing the endless. Going beyond, what you almost brush aside sets out the context of meaning for what follows. And no, I have no interest in the journals game. That effectively substitutes appeal to authority for fact and logic in the context where no authority will be better than facts and reasoning. The above is an elaboration on the challenge: claiming to have completed an inherently endless stepwise process. Fallacy name: I just completed the endless. Ooooooops. KF

  141. KF,

    DS, just to begin your manager would have to traverse a transfinite interval to reach a finite neighbourhood of 0.

    I would phrase it like this: given any natural number n, n seconds ago, the manager had already visited all but finitely many of the rooms.

    You’re not going to refute that using cardinal/ordinal arguments.

  142. I find it ironic that Kairosfocus, despite his name, appears to be unable to focus on a discussion. Here are 2300 more words saying, again, what he has said before, and that continue to address points his fellow discussants are NOT making while failing to address points that we are.

    kf writes,

    That is, there is a claim on the table of an infinite past that has somehow arrived at today.

    No. Neither dave or I have made that claim, or expressed interest in the topic of time. We have made it clear we are interested in the pure mathematics of the nature of infinity in respect to the natural numbers. I’ve even said you are right about the infinite past not being able to arrive at today.

    But you don’t seem to be able to grasp what is and isn’t on the table as a topic, and are thus unable to narrow the focus of your thoughts.

    kf writes,

    It is highly significant, in that context, to see the resistance and even dismissiveness in the thread above to the ordering pattern put up from
    0, 1, 2, . . . k, k +1 . . . w, w + 1, . . .

    No, Dave and I (after some explanation from Dave and further reading) have fully acknowledged the existence of the above sequence of ordinals.

    But that sequence is not a counting sequence relevant to the natural numbers: the above sequence is about something different.

    N = {0, 1, 2, . . . k, k +1 . . .} only.

    Again, kf keeps talking about something that goes beyond the topic

    kf writes,

    And, as there has been so much of foreclosing discussion by appealing to the finitude of “all” naturals, I have felt it wise to back away and go to the first principles of counting numbers and constructing ordinal sets instead. That is not evasion of focal issues, but instead trying to address them in a way that does not needlessly run into perceptual barriers.

    First principles are that every natural number k has a successor k + 1. Simple as that. For natural numbers, the cardinal value and the ordinal values are the same. Constructing ordinal sets as a first principle doesn’t help, because once you include w and its successors, you aren’t talking about the natural numbers any more.

    [Oops – I’d told myself I wasn’t going to counter-point anymore.

    Third and last point:

    kf continually makes points and uses terms that he doesn’t clearly explain: “going past the ellipsis” into “the transfinite zone” where “w kicks in” and “endlessness begins”. When we ask what these concepts mean and how they could apply to the natural numbers, there is no answer.

    As Dave said, and I agree with everything else Dave said also (I like the phrase “wrestling with fog”), kf should try writing up his ideas about the above vague notions in a way that a trained mathematician or journal could evaluate. He is claiming that a fundamentally accepted fact in number theory, that “all natural numbers are finite”, is wrong. If he could establish that in the world of mathematics, he would have some renown.

    I see that kf has added this:

    And no, I have no interest in the journals game. That effectively substitutes appeal to authority for fact and logic in the context where no authority will be better than facts and reasoning.

    That’s revealing. Who is to evaluate “facts and reasoning”, if not a body of people capable of understanding the arguments in the context of what is already known? To write a paper that clarified his thoughts and share it with mathematicians would be a way of getting feedback about whether his “facts and reasoning were sound.” In general, without such feedback, one can invent ideas in a vacuum and go on thinking one is right forever without accepting the challenge of convincing the larger world.

  143. To write a paper that clarified his thoughts and share it with mathematicians would be a way of getting feedback about whether his “facts and reasoning were sound.”

    You keep forgetting the primary fact: ‘KF is right and DS (and Aleta) is wrong. If math agrees with DS (and Aleta) then math is wrong, too.’

    That clearly tells you why writing a paper and getting it evaluated by a journal or other competent mathematicians is a completely irrelevant waste of time. Unless they agree whith KF (which they won’t), they are simply wrong as well.

  144. Each room is finite, but the hotel itself does not consist of a finite number of rooms.

    We can claim that each room has a room number, but who did the numbering, and when?

    Does the hotel have a 13th floor?

    Each floor has an infinite number of rooms, there are an infinite number of floors, and an infinite number of steps between each floor.

    Which room has the best view?

  145. Which room has the best view?

    The one at the top, of course! 😀

  146. But you have to go through the twilight … oops, transfinite, zone to get there! 🙂

  147. Aleta, I note to you that there is a substantial focus issue and that is what I have chosen to speak to as thread owner. KF

  148. ??? What do you mean? Do you mean that you want to talk about the “can’t get here from the infinite past” issue even though no one who is in the current conversation is interested? If so, fine, but then don’t bother to address me at the start of your post.

    Also, thread owner or not, very seldom does a 150 comment thread stay focused on the OP – usually one sub-issue, or a related issue, becomes the focus.

  149. Aleta, kindly read the original post, observing the two cites in it, thanks. KF

  150. Why?

    I’ve read them, several times.

    I HAVE NEVER BEEN DISCUSSING THE ARGUMENTS ABOUT THE NATURE OF TIME.

    I HAVE AGREED THAT “COUNTING UP” FROM NEGATIVE INFINITY IS IMPOSSIBLE.

    I HAVE JUST BEEN INTERESTED IN DISCUSSING THE NATURE OF THE NATURAL NUMBERS.

    Pardon my shouting, but why don’t you get those three points? I have repeated them a number of times.

  151. Aleta, the OP sets the focus for the thread. Your suggestions above that were personalised by use of my handle were inappropriate given the OP. KF

  152. I see – you didn’t like the “Kairosfocus can’t focus” comment.

    Yes, as the author of the OP you have the right to say that is what you want to discuss. In fact, anyone has the right to say what they do and do not want to discuss, as I did.

    However real discussions seldom stay on the original topic, and as a discussion evolves, the most important consideration to me is whether the people in the discussion are actually creating a constructive dialogue.

    One of the characteristics of a constructive dialogue is that people respond clearly and specifically to the other person’s points and answer the other person questions. In so doing, the topics and the participants’s respective positions gains clarity.

    This is what I mean by focus: staying clear about the course of the thread of a conversation, and participating in ways that create some clarity about the respective positions of the participants.

    This is what I was saying you were not doing. Of course you had the right to return the OP, and the notion of past time, if you wished. But given that you knew, or should have known, that the other participants in the current discussion were solely interested in the natural numbers and the analogical hotel, not time, perhaps you could have acknowledged that you were not in fact furthering the discussion with us.

    As a matter of general philosophy, having the right to do something and having it be a useful thing to do can be two different things. If you want to help create constructive discussions – ones in which all parties have a chance to both clarify their positions and gain greater understanding of the other positions, you should learn to focus, I think.

  153. I’d just like to say how pleasant daveS and Aleta have been compared to what we often see here at UD and to thank them.

    Interesting discussion.

  154. Thanks, Mung.

  155. Hilbert’s hotel reminds me of Ira’s Flophouse.

    Since it was pretty cold outside, three vagrants entered the derelict hotel. “How much for a room?” they asked the night manager. Removing the dead cigar stump from his mouth, the night manager said “30 bucks for all night.” Each of them coughed up $10.

    After they went off to their grimy but warm room, the manager came in. “How much did you charge them?” he asked. “30 bucks,” said the night manager. “Nah, I know those guys and promised them a room for only $25. Give them a $5 refund.” So as the night manager went to their room, he tried to figure out how to divide $5 three ways. He solved the problem by tipping himself $2 and refunding each of the vagrants $1. So, each of them had paid $9 for the room.

    Lessee. 3 x 9 = 27 plus the $2 that the night manager tipped himself comes to $29. What happened to the other dollar?

    -Q

  156. Folks

    I have a few moments.

    >>>>>>>>>>>>>

    Alicia & DS:

    Thanks for comments across the thread, and general tone is appreciated; as Mung has noted.

    The focal issue comes back to, it seems there is a fallacy to be addressed, claimed stepwise actual completion of the endless. That comes up in Spitzer, Durston and Craig, and from many other directions. Language itself is seemingly trying to tell us something here: finitude –> endedness, infinity –> end – LESS – ness. To claim completion of endlessness seems to reflect a category confusion.

    In looking at linked mathematics themes, the questions have come up on what it means to have a transfinite succession, and particularly what it means to have such in descending order to a finite neighbourhood of 0.

    I am not satisfied that pointing out that successive addition attains to natural number k then k + 1 adequately addresses the endless succession to arrive at transfinite cardinality.

    Something is missing.

    Something, tied to endlessness.

    Something, that then applies to claimed completion of a stepwise endless process such as the inspection of HGHI’s rooms from the far, transfinite span inwards to the reception area viewed as room 0.

    Where also, a reasonable symbolism and sequence model i/l/o the ordinality of the numbers that captures the transfinite span to be traversed is a significant issue. The line of successive ordinals is where that will have to be resolved, somehow.

    That is why I have sought to identify some ordinal of countably transfinite cardinality and address the issue of stepwise succession onwards down towards 0. Surely, if the past is endlessly remote, at some point it must have been transfinitely remote if such is so. So, what is a reasonable representation, and what does that tell us about stepwise traversal of the transfinite?

    And no, such a remote point (say even the much objected to A = w +g above) would not be a beginning of events, just a startpoint for counting downwards. (I sometimes got the impression above that there was an objection to the notion of carrying out a count that runs from the remote past; but such was once the present and at any present time we can start to count a progression: 1, 2, . . . and keep going. Obviously such will always run k, k + 1, etc, but that is the point. An inherently finite succession of steps seeking to span the endless seems inevitably futile. The misleadingly simple three dot ellipsis is covering a major process that implies a qualitative shift due to endlessness. The closest I have come is to suggest endlessness leads to an emergent phenomenon, the transfinite that comes in degrees, starting with the famed aleph null. And the interface with the ordinal succession to w and beyond is also significant. From this the issue of claiming of implying actual stepwise completion of the endless arises. For concrete example 1 + 1 + . . . 1 = k is a completed and finite process. 1, 2, . . . is not.)

    This has direct relevance to an underlying concern from OP on, on the claimed endless past of the cosmos and linked causal succession of stages to now.

    >>>>>>>>>>>>

    HRUN

    Pardon, but I must advert to earlier notes to you.

    It is clear that you have never registered that this area is the one where there was a major back-tracking on what is now termed naive set theory, that there are significant differences in thought among relevant professionals about actualised infinities and other themes.

    I suggest in future that you re-think projecting closed mindedness to others.

    >>>>>>>>>>>>>>

    Mung:

    Entertaining as ever.

    >>>>>>>>>>>>>

    Q:

    A good one.

    _____________

    Gotta run.

    KF

  157. Thanks for the reasonably succinct response, kf. I would like to respond to a few points. I’m going to try to keep each point on a specific question or comment in order to help separate the issues

    I have a favor to ask: would you be willing to respond to each point separately also? Perhaps we can sort out what the real issues are, and see if we can separate the common ground from the points of difference between us. I have bolded major questions that I would particularly like a short answer to.

    I am speaking for myself here, although I am pretty sure Dave and I are in agreement on the issues I’ll mention. Also, I am going to model the idea of responding clearly to each point by responding to each of your paragraphs.

    Anyway, I’ll try. Here it goes.

    1. You write,

    The focal issue comes back to, it seems there is a fallacy to be addressed, claimed stepwise actual completion of the endless.

    I am wondering who is making this claim?

    a) I am not claiming this is possible.

    b) the people making this claim, I think, are the people Spitzer et al are responding to: those who believe time has gone on endlessly from the past, with no beginning.

    But again, I am not making, nor defending, that claim.

    So are we clear that I, and I believe no one in this thread, has said a “stepwise actual completion of the endless” is possible.

    It seems to me we are all in agreement on this point.

    Do you agree that we are in agreement on the point that “a stepwise actual completion of the endless is impossible”?

    2. You write,

    In looking at linked mathematics themes, the questions have come up on what it means to have a transfinite succession, and particularly what it means to have such in descending order to a finite neighbourhood of 0.

    I have not been discussing a “descending order to a finite neighbourhood of 0.” We have, I think, been discussing counting up from 0, in discrete steps, and in discussing the nature of the endlessness that entails, but I have not been discussing “counting down” from infinity to a “finite neighbourhood of 0.” This is essentially the same point as a made in 1.

    Could we agree, then,

    a. that counting “down from infinity” is not a topic of discussion between you and me, and

    b. that the issue we are going to discuss, if we do continue to discuss, is the issue of counting up from 0, and thus the nature of infinity as it applies to the succession of natural numbers.

    If a is what you want to discuss, and not b, then it will be clear that we don’t have a common topic of interest, and so we can consider the discussion done.

    3. You write,

    I am not satisfied that pointing out that successive addition attains to natural number k then k + 1 adequately addresses the endless succession to arrive at transfinite cardinality.

    This seems to be a common point of interest.

    My position is that the natural numbers N = {0, 1, 2, … k, k + 1, …}

    a. are represented by the first order of transfinite cardinality aleph null, the number of numbers in the set N, and that w represents the corresponding first infinite ordinal. Thus aleph null and w are transfinite ideas associated with N.

    b. further transfinite orders of cardinality and ordinality are about other sets, not N.

    First, do you agree with 3a?. Second, do you agree with 3b?

    c. More specifically, in response to your statement of concern, can you describe what you mean by “the endless succession to arrive at transfinite cardinality.”, and specifically, what does “arrive at” mean. Can you flesh out your perception of what the issue is as we create the set of natural numbers by endlessly taking the step to the enxt natural numbers?

    4. The rest of your post is again about counting down from infinity, and about the possibility of an endless past. If your response to 3a and 3b above is that you want to continue to discuss counting down from infinity, and not the nature of the natural numbers as we traverse them going up from 0, then it is clear that we are not interested in the same topic.

    Being clear on that will be useful.

    However, if you are willing to accept that I agree counting down from infinity is impossible (see 1 above), and you still would like to discuss your concern about the natural numbers (see 3c above), then we can continue.

    I look forward to your response.

  158. KF,

    Surely, if the past is endlessly remote, at some point it must have been transfinitely remote if such is so.

    That’s not what I am assuming. To me, the existence of an infinite past just means that given any natural number n, the universe already existed n seconds ago.

    [edited] This is consistent with every instant in the past being a finite number of seconds away from the present.

    Mung: Thanks.

  159. Dave:
    That’s not what I am assuming. To me, the existence of an infinite past just means that given any natural number n, the universe already existed n seconds ago.

    [edited] This is consistent with every instant in the past being a finite number of seconds away from the present.

    I think KF has it backwards. You don’t move from the past to the present , every moment in the past is the” present ” relative to that moment. You are always at ( 0 ) on the line. It is the configurations of space that are the points on the line. In that sense, a possible infinite number of configurations makes the configuration that we term the present not impossible but probable.

  160. DS, so, 13.75 bn ya, the observed cosmos was, and 10^500 y past and . . . [that transfinite ellipsis again]. KF

    PS: For the moment I have to have focal effort elsewhere, so overnight.

  161. KF,

    Eh? I’m not sure what you’re saying there.

    [Edit: If you’re saying (to summarize my statement above) that 13.75 billion years ago, the universe already existed, and 10^500 years ago, the universe already existed, and likewise for any finite number of years ago, then that is correct.]

    All the ellipsis means in this context is that the set is closed under the successor operation.

    When we write {0, 1, 2, …}, we’re saying that the set has 0 (and 1 and 2) as elements along with the successor of any element. It’s nothing terribly deep.

  162. I wasn’t under the impression that our universe having an infinite past was ever the issue. I think everyone here, including kf, Spitzer, etc. accept that our universe started 14 billion years or so ago., so that time within our universe definitely has a finite past.

    My impression has been that we have been assuming that time in some universal Cartesian sense stretches back before this universe – the very first post in this series of threads mentions a series of universes.

    And Spitzer’s original arguments are about an abstract, mathematical infinite past, not past as embedded in this universe.

    So, to kf, I want to make it clear that my post at 159 has nothing to do with the issue of the age of the universe, or the existence of other universes, or anything related to that.

    So my hope is that you would make at least one response to 159 without getting these other issues involved.

  163. Aleta, passing by again a moment. There are people out there claiming an endless past; which I have been representing as before singularity at 0 — note OP ” some naturalists such as Sean Carroll suggest that all we need to do is build a successful mathematical model of the universe where time t runs from minus infinity to positive infinity”. DS, in saying at any n in the past there was a world is implying just such an endless past. 1, 2, . . . n . . . rinse and repeat. That is where my concern on traversing the transfinite in steps comes from. KF

  164. When you have more time, I hope you can look specifically at 159 and respond to my points so as to keep them separate from other points.

  165. KF,

    DS, in saying at any n in the past there was a world is implying just such an endless past. 1, 2, . . . n . . . rinse and repeat.

    Clarification: What I am saying is that an infinite past does not entail the existence of any points in time infinitely far in the past.

  166. Still busy, later.

  167. Aleta:

    Having addressed a local issue, pardon my having had to be basically offline, I will pick up from 159:

    1: >> I am wondering who is making this claim? [claimed stepwise actual completion of the endless]>>

    Asa I briefly noted, that is one of the implications of claiming an infinite past, which has not only been asserted out there, but in thread, DS has gone on to say, 160: “To me, the existence of an infinite past just means that given any natural number n, the universe already existed n seconds ago. [edited] This is consistent with every instant in the past being a finite number of seconds away from the present.”

    So, it seems to me the claim is at minimum implied and is converted into an attempt to say that any particular past moment will be finitely remote so there is no descent from the transfinitely remote past. And, by extension of causal succession, there is no transfinitely remote distant past just at any given point, finitely remote values, which correspond to numbers.

    I believe this claim is patently false, if a transfinite past is claimed, a transfinite causal succession is claimed and an achieved completion of the endless is claimed.

    Which is so obviously problematic that here is an attempt to suggest that everything within the series, n, is finite but the whole is transfinite.

    A transfinite past succession and an endless past train of steps thus in principle numbers to go with it, is there.

    And, on fair comment, it does fall under the stricture of trying to claim or imply completion of the endless.

    You are not making such a claim (as we both acknowledged long since), but others have or strongly seem to me to imply such.

    2: >>I have not been discussing a “descending order to a finite neighbourhood of 0.” We have, I think, been discussing counting up from 0, in discrete steps, and in discussing the nature of the endlessness that entails, but I have not been discussing “counting down” from infinity to a “finite neighbourhood of 0.” This is essentially the same point as a made in 1.>>

    Again you may not have, but the issue is there in context, given what has been at stake.

    My concern in part has been, how can we represent such a claim symbolically, on the way to understanding it.

    I think the best we can see so far is that ascending count from 0 or a finite neighbourhood of that in an attempt to attain w etc is inherently futile, w is in effect an emergent value once endlessness of succession is in play.

    We can only actually ascend in steps to a finite point, but the issue of endlessness is real. It then gives rise tot he issue that if there is an implied endlessness of descent, it will be challenged to complete in a finite neighbourhood of 0.

    It seems to me the two issues of going up or down are entangled, as the spanning of the transfinite in steps is implied in both.

    3: >>My position is that the natural numbers N = {0, 1, 2, … k, k + 1, …}

    · a. are represented by the first order of transfinite cardinality aleph null, the number of numbers in the set N, and that w represents the corresponding first infinite ordinal. Thus aleph null and w are transfinite ideas associated with N.

    · b. further transfinite orders of cardinality and ordinality are about other sets, not N.>>

    My concern remains, that N — which contains the counting numbers as extended in order endlessly — as a set has transfinite cardinality, and the weight of that should not in effect be left to a three dot ellipsis.

    I am even uncomfortable with the argument If k is finite and k + 1 is its successor, then any pair k, k + 1 will be finite, and 1 is finite so all numbers in succession thereafter are finite. The problem being, the matter in view to discuss successions is exactly dependent on finite succession when the ellipsis points to endlessness. I think we are here close to begging a question or two, uncomfortably close for me.

    How do I put it.

    Something like, a finite increment to a finite is a finite indeed, however we are arguing to the endless and in a context where an endlessness is transfinite. So, a transfinite number of finites where the very numbers themselves are what is in view to attain to cardinalities, sits uneasily for me.

    Perhaps, I am being overly scrupulous or needlessly concerned but it will not shake.

    4: >>The rest of your post is again about counting down from infinity, and about the possibility of an endless past.>>

    Yes, that seems entangled.

    KF

    PS: White screen of death again.

  168. Thanks, kf.

    I will limit my comments those pertaining to the natural numbers.

    Tu write,

    Perhaps, I am being overly scrupulous or needlessly concerned but it will not shake.

    Yes, I think you are. The key to what I see as your confusion, and the source of your concern, is contained here:

    I think the best we can see so far is that ascending count from 0 or a finite neighbourhood of that in an attempt to attain w etc is inherently futile, w is in effect an emergent value once endlessness of succession is in play.

    An ascending count is not an “an attempt to attain w.” w could be considered an emergent value associated with N, being the ordinal number associated wth aleph null, the transfinite number defined as the order of infinity represented by the natural numbers.

    But an ascending count doesn’t “attain” anything: it just goes on and on because there is nothing to ever cause it to stop. Every k is followed by k + 1. It’s not trying to get to some other level of number w, so there is no futility involved. w is a number about N, but it is not a destination of some kind within N.

    And then you write,

    We can only actually ascend in steps to a finite point, but the issue of endlessness is real. It then gives rise tot he issue that if there is an implied endlessness of descent, it will be challenged to complete in a finite neighbourhood of 0.

    It seems to me the two issues of going up or down are entangled, as the spanning of the transfinite in steps is implied in both.

    No, there is no “spanning of the transfinite” in counting up. Nothing is ever spanned, the ellipsis is never passed. As I said above, the transfinite is about the whole set of natural numbers, but not about some place “beyond the ellipsis” within the natural numbers.

    And since there is no transfinite spanning counting up, there is no transfinite spanning “counting down”. “Counting down from infinity” is impossible not because a transfinite span must be completed, but because there is no such place as “infinity” to start counting down from. Wherever you started counting down from will be a finite number, and you could always start counting down from a higher number.

    So, this is the heart of the matter, I think.

    Thanks for the response. I’m pretty sure this is all I have to say: I don’t think there is anything more I could say. I think you have intuitive concerns about the nature of infinity – it’s a baffling subject, but I do think you have an erroneous concern about there being some unresolved mystery connected with those three little dots …

  169. KF,

    Suppose the universe were spatially infinite, which I assume you agree is at least conceivable.

    Would that then mean that there must exist points in the universe infinitely far from Earth?

  170. I don’t agree with the concept of minus infinity to positive infinity. It leads to stupid results. For Eg,
    Most particles have a lifetime of microsecond. If they start moving near speed of light, their lifetime increases.In ‘0 to negative infinity’ time belt,if they start moving, their lifetime will decrease! In fact, because lifetimes are in microseconds for most particles, in the ‘0 to negative infinite’ time belt, particles will cease to exist if they move!

  171. FWIW, one late correction to my post #104: The ordinals do not form a set, but rather a proper class.

  172. Aleta,

    I just woke up with the infinite on mind, and thought to look here.

    I think a contrasting case will help. The interval [0,1] as a continuum has in it transfinitely many real numbers, and to complete the process of say travelling across one metre traverses a transfinite succession of such a continuum in a finite time.

    For this, there is no problem, these are processes within finite limits. And of course a translation can then extend any continuum between [0,1] to any span in stepwise succession, so the notion of a continuous line, then plane then space then hyperspace is no problem.

    Now, we go to endlessness, and to the issue of assigning the value aleph null as cardinality to an endless succession of counting numbers considered as a set. First order of magnitude endlessness, and then one may assign onward values by a succession of power sets.

    Further, one may define w as the successor to the process as imagined to continue endlessly, and go on from there to w + 1 etc, on to epsilon nought etc.

    In context, a prime issue is mathematical induction, seen as chaining onwards, where if X is so for initial member i and the logic of succession is that if X(K) then X(K+1) this chains onwards in ordinal succession endlessly. As opposed to as a complete in fact process.

    And, I confess to getting just a tad concerned when there seems to be a hestitation to look at coming back down once one looks at the transfinite range of ordinals and cardinals. If simply changing direction of succession and start-point can be so sensitive, that is not a healthy sign.

    Beyond, it is clear that one cannot actually complete an endless stepwise process, the stepwise process is inevitably potentially but not actually infinite.

    So, to premise a completed process is already to move to the world of what is potential and conceptual, not physically actualised, as a general rule.

    Going back to the span [0,1] I am seeing that again the traversal is conceptual, we do not actually work out the endless succession to arrive at a given point, say 1/pi in that succession, or any one of the endless chain 1/ (pi^n) which we can conceptually catapult into the zone, we note that we can conceive of the continuum, and use it to model the actual world. Between any two neighbouring but distinct values, we may in principle define a third, most easily by an averaging process. (BTW, I can sympathise with Mapou in his discarding of the infinite, though I think there is a legitimacy to the conceptual space and to its mapping of the actual world.)

    Now, let us look at your onward remarks.

    Let us see what I spoke to in your first clip from me in 169:

    ascending count from 0 or a finite neighbourhood of that in an attempt to attain w etc is inherently futile, w is in effect an emergent value once endlessness of succession is in play.

    In short, in the first part, I am saying the completion of endlessness is a futility, a supertask. I am not envisioning this as actually done. And, I am seeing w as the concetualised successor to that process, and as projected once endlessness of succession of steps is in process.

    Likewise, the point of endlessness is it implies a spanning of a transfinite range at least as a concept, and further entails that such is impossible as a result of a stepwise finite process. To span, we must catapult across the range conceptually, we need a mathematical wormhole. That seems to be in part what the use of y = 1/x to discuss infinitesimals and hyper reals is about. As you recall, I have discussed what I can now say is a “mild” infinitesimal m, that catapults you [conceptually . . . ] into the range of w, at w + g, g finite. Where m in my view is part of [0,1] so I have no reason to hold it not a valid conception, and I accept the existence of hard transfinites h — now, that is a happy coincidence that goes to the classic first principles of differentiation: f'(x) = lim as h –> 0 of {[f(x +h) – f(x)]/h} — in said range and close neighbourhood of 0.

    Forgive the sci fi terminology.

    H’mm, let me add to this.

    As between any two values in [0,1] we may define a third, we see that we can identify that if m and n are neighbouring mild infinitesimals, then their catapult values will be just as near, A, B, i.e. we see that there can be a line of continuum defined in between successive ordinals w +g and w +h, say. (I here use h in a different sense, sorry.) Where of course between m and n we can interpose an intervening neighbour and catapult that too.

    That sure looks like, transfinite continuum.

    Which would also perforce extend tot he hard infinitesimals, so the hyper reals would be continuous, of course involving hyper integer values among them.

    And, I see that once non standard analysis is on the table, infinitesimals are back in business. So, it makes sense to speak of mild ones. And to apply a function to transform x-value at input to y-value at output is a reasonable process. Here, in the very near neighbourhood of 0 for x.

    Coming back to the next series of your remarks:

    No, there is no “spanning of the transfinite” in counting up. Nothing is ever spanned, the ellipsis is never passed. As I said above, the transfinite is about the whole set of natural numbers, but not about some place “beyond the ellipsis” within the natural numbers.

    And since there is no transfinite spanning counting up, there is no transfinite spanning “counting down”. “Counting down from infinity” is impossible not because a transfinite span must be completed, but because there is no such place as “infinity” to start counting down from. Wherever you started counting down from will be a finite number, and you could always start counting down from a higher number.

    yes, we agree no stepwise finite step process will span a transfinite zone. Hence my starting from the idea of a catapult via 1/x from what I would now term a mild infinitesimal, m. And, numbers inherently are a conceptual space that has a conceptual span, though obviously a transfinite one.

    I would suggest that once we have the zone, w, w +1, w +2 . . . w +g . . . w + w . . ., we can and do have up/down successions in a transfinite band. So, it makes reasonable sense to do a down count from some w +g, g finite and ask, what happens if we keep on endlessly? To which the answer is, we cannot escape the transfinite zone by stepwise succession downwards that actually attains the finite neighbourhood of 0, the cardinality is still of order aleph null. At least, that seems reasonable so far.

    And no, it seems to me that a succession upwards or downwards in a transfinite range of numbers is reasonable as they can be laid out in order of ascent and simple request, what, sir is your predecessor and a pointer to the left of one step will succeed. Until one hits a transfinite ellipsis of endlessness.

    w + g, w + (g – 1), . . .

    or, w +g [= A], A ~1, A ~2, . . .

    [Read, A less one, etc]

    Where such are obviously very powerful symbols in this business we are attempting here.

    Let me add to the set of symbols, to symbolise endless continuation with: . . . EoE . . .

    Adapting Wolfram:

    From the definition of ordinal comparison, it follows that the ordinal numbers are a well ordered set. In order of increasing size, the ordinal numbers are

    0, 1, 2,. . . EoE . . . , omega, omega+1, omega+2, . . . EoE . . ., omega+omega, omega+omega+1, . . . EoE . . . [then full stop].

    The notation of ordinal numbers can be a bit counterintuitive, e.g., even though 1+omega=omega, omega+1>omega. The cardinal number of the set of countable ordinal numbers is denoted aleph_1 (aleph-1) [and, it corresponds to w1].

    Symbolising, it seems we are looking at:

    . . . EoE . . . w+g [= A], A ~1, A ~2, . . . w . . . EoE . . . r, (r-1) . . . 2, 1, 0.

    [r defining a convenient finite neighbourhood of 0]

    Now, we can say, the EoE cannot be spanned in steps. And any stepwise process will be finitely remote from its start point at any actually completed stages k, k +1 and so forth, repeat.

    In this context I can see holding, EoE is a roadblock to finite process, or rather an un-span-able Sahara. You run out of resources long before you can span it in steps.

    Now you speak of how ” . . . the transfinite is about the whole set of natural numbers, but not about some place “beyond the ellipsis” within the natural numbers.”

    Of course, we have the succession conveniently provided by Wolfram, and the implication of successive steps being that what we can actually count to will be finite. But there is endlessness so the span of the whole, as the potential and abstract endless process is transfinite. An emergent conceptual property of the whole.

    Fine so far.

    Howbeit, at a price. Endlessness within the counting succession is involved inside the set, via the ellipsis again:

    { 0, 1, 2 . . . EoE . . . }

    And, we are here dealing with the very set used to count.

    Its span MUST be endless, transfinite.

    So, while any actual succession of actual steps must be finite, the span of the whole as we may abstract from taking steps and applying succession, is transfinite.

    Where, too, it matters not that we define w as NOT a “natural” as the EoE is inside the set of naturals, I have just made that span explicit. The naturals keep going on and on ENDLESSLY. We can only attain to finite degree but the EoE says, onwards forever. And endlessness is the very stuff of the transfinite in mathematics.

    So, further, we see that the claim to complete a downwards succession within that EoE to a finite neighbourhood of 0 is a futility.

    And, to assert all naturals by succession and induction are finite, is to imply spanning the transfinite. I am thinking, we modify that any completed span will be finite. We may only actually point conceptually to the whole. Which is valid as mathematics is a conceptual exercise in the first place.

    So, yes any actual count that attains a finite neighbourhood of 0 in the downwards direction will be finite.

    Which is the same as saying, the actual succession will be finite in cumulative span, the EoE cannot be bridged in actually completed steps.

    Which for my interest, has direct relevance to a claimed past transfinite causal succession, or at any rate, one that has to face EoE.

    And yes, the three little dots do point to something awesome and mysterious.

    Onwards, with the explicit acknowledgement of the EoE in the set of counting numbers — which identifies where counting can potentially but not actually range to — I can see that any natural or counting number we can attain is finite, but the set as a whole has EoE in it, and that gives it transfinite character.

    And yes there is a distinction I make there, that points to the concern I have on asserting all natural numbers are finite. All values or members we can attain to by a stepwise successive process are finite, but the whole set contains EoE so is of inherently transfinite character. And we may proceed to define a successor to that EoE, w, and then succeed it w + 1, etc, within a range that starts in the transfinite and extends the concepts.

    That, I am comfortable with.

    KF

  173. DS, Endless extension in actual space is just as problematic as endless extension in time. KF

  174. MT, I hear your concerns. I guess this is the thread for concerns — I have long meant to add stir the pot as a category, I will do so in a moment. KF

  175. F/N: I see we are in the high hit for the past month club here now.

    Any new participants, please understand this is an exploration, live, messy, incomplete, patently vulnerable.

    I am uncomfortable with two things, claiming an infinite down-count that gets to a finite neighbourhood of 0, and the claim that all naturals are finite but the set of naturals is transfinite.

    As at 174, I am coming to a point of comfort by introducing an explicit ellipsis of endlessness and holding that it is INSIDE the definition of the set of naturals.

    So, ruling that w etc are not naturals is not relevant to the main concerns.

    And, I am seeing that we can suggest a catapult — the function y = 1/x — to get us from [0, 1] the closed and continuous real interval, to the zone by way of a process at least conceptually comparable to how discussion of hyper-reals and infinitesimals is entertained in non standard analysis.

    All of this then extends tot he issue of infinite stepwise succession on finite stages, and to the claim or implication that here is an actual infinite past to the physical, matter-energy space-time cosmos and extensions thereof in some form or other.

    Finite successive causally connected stages would be on the table an these can be labelled in succession

    . . . C2, c1, c0 | . . . singularity | C1*, C2*, . . . Cn*

    Where Cn* is now.

    The math is directly logically connected.

    I argue that claiming or implying completion of an endless successive process of finite stages is a futility, a supertask and so the best conclusion is our cosmos etc are of finite temporal span, had a beginning at some finite time, including whatever physical may lurk behind the singularity of what 13.75 BYA on the usual timeline.

    And to pop a physical world out of a nonexistent hat of non-being at some point is even more of an absurdity.

    Mix in ontological issues on necessary being foundational tot he actual existence of any world, and we are staring eternity — as opposed to time — in the face, folks.

    At least, that is how it looks to me.

    So, if you have thoughts, welcome.

    KF

    PS: Those who wish to cynically dismiss me as having fixed and unalterable ideas, this thread is in part a demonstration of the opposite, it is of exploratory character in the context of an intuitive sense of discomfort with common claims. Why that discomfort, apart from there is some incoherence somewhere, there is some circle of begged questions or both, or, what? I want to be at a point of comfort or at least lessened discomfort due to having had a serious open exploration of the issue.

    PPS: If you thing the ordinal chaining borrowed from Wolfram for convenience and extended as well as issues of down counting etc tied to such not to mention catapulting from [0,1] to a transfinite range are all wet, kindly, show why.

  176. kf. I appreciate your characterization of this discussion as an exploration: I think that is what good constructive dialog ought to be. However, I’m finished with my part of the discussion: trying to address and make sense of the many points you bring up, many of which I’ve already said I’m not interested in, would not be a good use of my own time and energy. You may be uncomfortable with “the claim that all naturals are finite but the set of naturals is transfinite,” but I’m not, so I’m ready to let he discussion come to an end.

  177. KF,

    And, to assert all naturals by succession and induction are finite, is to imply spanning the transfinite. I am thinking, we modify that any completed span will be finite. We may only actually point conceptually to the whole. Which is valid as mathematics is a conceptual exercise in the first place.

    Most mathematicians don’t think of the natural numbers as a being in a partial state of completion. Rather, the set N is already “completed”, if you will.

    Can you give a good reason for not going ahead and flatly stating that all natural numbers are finite, period? Is it possible that we will at some point discover an exception?

  178. DS, look at the set itself: the counting numbers, which is by definition endless and would contain all numbers ordered from 0, 1, 2 . . . EoE . . . I am looking instead at a definition, N is the least inductive set (set of successors to 0, 1 etc in effect). KF

  179. Ok, I don’t think I’m seeing the problem, however.

  180. The … already means endless continuation. What do you gain by adding another symbol EoE to also mean endless continuation? It all means the same thing – there is always another finite integer.

  181. You writem “Of course, we have the succession conveniently provided by Wolfram,”

    The “the succession conveniently provided by Wolfram” is not all within the natural numbers:

    N = {1, 2, 3, …}

    N does not equal {1, 2, 3, … w, …}

    In the interest of clarity: Do you agree that N does not contain w? Yes or No?

  182. Aleta,

    First the ellipsis is ambiguous.

    Second w is successor to the naturals, per the usual understanding.

    The issue is on the subject, endlessness is within the naturals.

    KF

  183. There is nothing ambiguous about the …. All it means it that for any k, k + 1 is also a natural number. That is all it means.

    N = {1, 2, 3, …}

    N does not equal {1, 2, 3, … w, …}

    In the interest of clarity: Do you agree that N does not contain w? Yes or No?

  184. Aleta, I already gave the answer: w is successor to the naturals, which has a meaning in ordinal context that reflects taking in what is before then capping it. Elsewhere, I spoke of the issue that the endlessness is already in the naturals, expressed in the EoE. But that’s the problem/point right there, the endlessness is in the naturals. KF

  185. I don’t get it, kf. I don’t see the problem. I don’t see that you have answered my question, either.

    Is w in the set of natural numbers or not? Saying “w is the successor to the naturals” doesn’t answer the question – is it a successor in the naturals, after the ellipsis, or is it a successor to the naturals – beyond but not in the naturals.

    Which is it? Why won’t you/can’t you say?

  186. Aleta this is an endless loop, I have repeatedly answered the question, in the negative. However the further material point, is that endlessness is in the set of whole counting numbers itself and w is in effect its successor as a whole, a value assigned to the first degree of endlessness as an ordinal. So, w is not an arbitrary, unrelated imposition which is exactly why the sequence shows it as successor to the whole counting numbers as they amount to endlessness. That is there is an organic connexion and a phenomenon of emergence to be recognised, endlessness. Endlessness that is within the set of counting numbers, where succession is such that, e.g. 5 –> {0, 1,2,3,4} and so forth, i.e. counting sets if you please have successors. We can then reasonably ask, go to the point of endlessness and ask, what is the onward successor, and that is assigned w. KF

  187. OK, that is clear: w is the ordinal successor to the natural numbers, but is not a natural number.

    Then what is wrong with saying every natural number is finite?

  188. … I … can’t … seem … to … stop … counting …

    Mr. Escher, we have a problem.

  189. Cleanup on aisle … EoE … k + 1 please!

  190. In fact, kf, if w (or the corresponding aleph null) are the first transfinite numbers, and they are not in N, then that proves that all natural numbers are finite.

    Q.E.D.

  191. Aleta,

    I wish it were that simple, and I am fully aware that in this mix is the mathematical proof by induction that I first learned to use in 6th form math a long time ago now.

    The matters at stake here go to not only mathematics but meta issues, phil of math issues, nature of sets etc.

    The key point is, we are dealing with counting sets and the meaning of numbers, e.g.:

    {} –> 0,

    {0} –> 1

    . . .

    {0, 1, 2, 3, 4} –> 5

    Indeed many would dispense with arrows of assignment and simply use equality of definition.

    I add: the collection, assignment process is what we effectively mean when we say 5 is successor to 4. And cardinality of 5 emerges as being an equivalence class, i.e. the set is a five-set and can be put in 1:1 correspondence with any other such. That is, we use an example that in this respect is WLOG. Onward, sets of countable transfinite cardinality will be such that proper, limited subsets can be put in 1:1 match with the unlimited succession of counting numbers; famously the evens and the odds.

    . . .

    {0, 1, 2, . . . EoE . . . |} –> w

    That is, w as successor is inextricably entangled with what has gone before.

    The ellipsis of endlessness is INSIDE the set of all counting numbers, it is “just” an assignment that this counting set is termed w. We cannot cleave w apart from what has gone before, it is entangled into what w means.

    And, obviously, to go, 1 + 1 + . . . 1 k times –> k and then one more to exceed it as k + 1 showing k is bound and finite, does not remove the significance of the EoE.

    Ellipsis of endlessness.

    Counting is not separable from the succession of counting sets.

    The endless list of such sets.

    Let me add: we are counting (in principle) with these sets and that requires an endless chain of endlessly increasing members.

    So, we see that we have a potentially infinite succession, a limitless process of counting sets, and when we consider that endlessness as a whole– one, that we may imagine but cannot complete in steps on pain of trying to end the endless — we say the successor is w.

    For induction, setting claim C is so for initial value

    C_0 or C_1

    and it chains as

    C_k => C_k+1

    simply embeds that potential infinity, the “it is so without limit,” the subscripts imply the presence of the endless succession of counting sets.

    To then say, voila, QED, all counting sets in that endless succession are inherently finite runs into, the endlessness.

    We are open ended, unlimited and endless thus not bounded at the upper end, so to claim or imply that all counting sets are finite runs into trouble.

    The unlimited by definition cannot be limited.

    You cannot end the endless.

    So, I think there is need to rethink.

    At least, for me.

    This is close to the heart of my discomforts, my sense of cognitive dissonance.

    I think the chaining, the linked axiom of infinity or whatever (and yes I know the independence issue obtains as with the axiom of choice and whether c = aleph 1) are telling us something that needs to be hedged around carefully.

    The EoE puts endlessness into the chain of counting sets, tantamount to the ordered succession of natural numbers.

    On an induction argument, what is proved — strictly — is, this succession of results is endless and reliable. It is an endless chain of stepwise logical transfer with good links hanging from an initial demonstrated mathematical fact.

    But for it to be valid itself, it cannot be bound, it must succeed itself in an unlimited chain as the subscript goes on forever increasing.

    Chain out in k steps to k and you can go on to k + 1, for any value you please.

    Which means, the set of successive counting sets has to be open to limitless extension. To assure that very reliability.

    The very opposite of finiteness.

    Yes, any particular, specific counting set value we may assign k can be exceeded k + 1, but it is looking a lot like a fallacy of composition to use that inherently finite point to bind the set as a whole when its essence is endlessness.

    So, the endlessness is in the set, not in w as designated successor, as w is composed from, emerges from, is inextricably entangled with that endlessness of succession WITHIN the set. Indeed w MEANS that.

    That entanglement is the root of my concern.

    I am comfortable in accepting chaining as inherently unlimited, but that simply further underscores the point.

    I add: once the set of counting sets is endless it has in it members that cannot be reached by a finite successive process, i.e. it is actually endless and beyond step by step exhaustion, on pain of not being endless. We can look on and point to the endlessness but cannot reach it and no k so k + 1 — inherently finite — will reach the extreme zone. Yes, by appending yet another EoE to the k, k +1 succession (notice how neatly we embed yet another endless counting chain on a proper subset by in effect using a start count from k and proceed to match k –> 0, k + 1 –> 1 etc . . . showing the transfinite nature here by 1:1 correspondence with the original, full set; and in fact we could make endlessly many match-able copies like that . . . ), we may POINT to the endlessness, but we do not actually attain it, and if we did it would no longer be endless, a contradiction would have occurred: ending the endless. I have spoken of an impassably vast Sahara and how we need to catapult past it using something like y = 1/x as we approach 0 in the [0,1] interval, using continuity and appealing to infinitesimals. At least that is how I am thinking.

    Where, further adding: yes, shifting to reals, I am also looking at the all but zero small, the infinitely small. I find no reason to reject that a mild infinitesimal, m can catapult us into a zone finitely near w, even as we may speak of hard ones that catapult us into hyper-reals beyond all reals as it is roughly suggested. And the continuum of [0,1] would then allow filling in the gap between w + g and W + (g + 1) by suitable extension of the number system model, however informally I have argued. That is, I can see the reals or hyper reals if you will filling in between the transfinite ordinals endlessly too; there is plenty of room at the bottom near 0 as between any two neighbouring values m and n down there, there will be another, say p. So, we can catapult the [0, 1] continuum between any two successors. I see endlessness of particular values at the bottom among the reals in a limited range continuum, and endlessness at the top too including endless fitting in of the continuum by such a catapult process, between successive transfinite ordinal values say w + r, w + (r + 1).

    Mathematical induction establishes unlimited reliability and endlessness, it does not stamp finitude into the set of counting numbers.

    On pain of undermining its own reliability.

    I add: And the notion that we have an endless number of FINITE values exhausting (in principle?) the succession of counting sets — what we use to count and to show endlessness itself — is very, very jarring here.

    Where, endlessness cannot be severed from the members, a set being, roughly, a definable collection.

    This is now beginning to bring out more of the force of my concern.

    KF

  192. F/N: Wolfram on limit ordinals:

    Limit Ordinal

    An ordinal number alpha>0 is called a limit ordinal iff it has no immediate predecessor, i.e., if there is no ordinal number beta such that beta+1=alpha (Ciesielski 1997, p. 46; Moore 1982, p. 60; Rubin 1967, p. 182; Suppes 1972, p. 196). The first limit ordinal is omega.

    KF

  193. F/N: Wolfram on transfinite induction:

    Transfinite induction, like regular induction, is used to show a property P(n) holds for all numbers n. The essential difference is that regular induction is restricted to the natural numbers Z^*, which are precisely the finite ordinal numbers. The normal inductive step of deriving P(n+1) from P(n) can fail due to limit ordinals.

    Let A be a well ordered set and let P(x) be a proposition with domain A. A proof by transfinite induction uses the following steps (Gleason 1991, Hajnal 1999):

    1. Demonstrate P(0) is true.

    2. Assume P(b) is true for all b<a.

    3. Prove P(a), using the assumption in (2).

    4. Then P(a) is true for all a in A.

    To prove various results in point-set topology, Cantor developed the first transfinite induction methods in the 1880s. Zermelo (1904) extended Cantor's method with a "proof that every set can be well-ordered," which became the axiom of choice or Zorn's Lemma (Johnstone 1987). Transfinite induction and Zorn's lemma are often used interchangeably (Reid 1995), or are strongly linked (Beachy 1999). Hausdorff (1906) was the first to explicitly name transfinite induction (Grattan-Guinness 2001).

    Where also:

    Principle of Weak Induction

    Let D be a subset of the nonnegative integers Z^* with the properties that (1) the integer 0 is in D and (2) any time that n is in D, one can show that n+1 is also in D. Under these conditions, D=Z^*.

    These first point to limit ordinals, where w is the first, as posing a logical barrier necessitating going beyond finite chaining exactly because of the EoE effect.

    So, indeed, we can define the counting numbers as those in principle reachable by successive logical chaining from 0 or 1. Then we can point to the endlessness involved.

    But, but, but we see here that the naturals are viewed as finite.

    I suggest, rather, we see the distinction between reachable in principle and endlessness. The number w is a systemic succession not a particular value in a chain, succeeding from some value k such that k +1 = w, it is a limit ordinal.

    Endlessness of succession is embedded in its meaning.

    I again point to the need to take that endlessness seriously.

    I would modify the above accordingly.

    The naturals are those counting set numbers in principle reachable by stepwise sucession from {} –> 0, {0} –> 1, {0,1} –> 2 etc, noting that they are endless in succession. And here I am using definition in a conceptual, even philosophical sense, fishing for the inherent, apt description of the essential meaning of a phenomenon, i.e. I am feeling for the innate nature of these numbers not imposing an arbitrary axiom that then controls what the meaning is. That endlessness in the definition — amplified from the ellipsis — implies that no stepwise process can exhaust them. Finite stage stepwise induction from initial case C_0 and C_k => C_k + 1 will point to but cannot stepwise exhaust the endless set. Where such a proof then hangs an unlimited reliable chain from a first demonstrated mathematical fact.

    That seems to capture what the set of successive counting sets or natural numbers is about, at least as I can see it just now.

    Then w succeeds, not by increment of unity k + 1 *=* w . . . NOT, but by being defined as successor to the endless set.

    Where also, it seems reasonable to use mild infinitesimals near 0 in the continuum [0,1] to catapult to the transfinite zone by the use of y = 1/x. From this, it further seems that between w and w + 1 etc, one may catapult a continuous zone thus filling in a line we may suggest as a trans-real line by analogy with the hyper reals.

    And implicit in this would be model assumptions and a sort of chaining by inductive succession.

    The question is, would such be reasonable as giving a finer ordering?

    There is a lower bound w [a limit ordinal], and there is a catapult mechanism that exploits the [0,1] continuum in the all but 0 lower end, and as if m >n, 1/m < 1/n, we can assign succession and we can also see that there is always a p between in the [0,1] continuum, so it seems reasonable.

    So, maybe here is a zone worth expanding on.

    And of course I am here using the definition that mathematics is logical reasoning about structure and quantity (which implies inter alia, sets).

    KF

  194. All more of the same, kf, without anything but vague concerns and without addressing, among other things, my simple point in 192.

    You write,

    I add: once the set of counting sets is endless it has in it members that cannot be reached by a finite successive process, i.e. it is actually endless and beyond step by step exhaustion, on pain of not being endless. We can look on and point to the endlessness but cannot reach it and no k so k + 1 — inherently finite — will reach the extreme zone. …

    And the notion that we have an endless number of FINITE values exhausting (in principle?) the succession of counting sets — what we use to count and to show endlessness itself — is very, very jarring here.

    Yes, the set is infinite, and no amount of steps can get to the end of it, because there is no end. But at each step you are at a finite number – there are just an infinite number of finite numbers.

    You keep using phrases which betray a deep misunderstanding – you can’t “reach” endlessness, and there is no “extreme zone” within the natural numbers. It’s “inherently futile” to think about “reaching the extreme zone” because the idea itself is erroneous.

    So back to 192: if all known transfinite numbers (cardinal and ordinal) are not actually members of N, then does it not follow that all numbers in N are finite? What alternaives are there? Either all natural numbers are finite, or there are transfinite numbers in N. If we have eliminated the second possibility, the first statement is true.

    “Jarring” concerns about an endless number of finite numbers may express a natural (which is sort of a double entendre, I guess) sense of the mystery of infinity, one which Cantor et al plumbed for us. But unless you can actually establish something mathematical (which my post at 192 highlights), your sense of being jarred is waiting for you to come to terms with that mystery.

  195. KF,

    Mathematical induction establishes unlimited reliability and endlessness, it does not stamp finitude into the set of counting numbers.

    It seems to me if you reject the inductive proof that all natural numbers are finite, then you reject the validity of mathematical induction in general.

    And since N itself is generally defined inductively/recursively, I don’t see how you make sense of the existence of N as a set itself.

    Yes, any particular, specific counting set value we may assign k can be exceeded k + 1, but it is looking a lot like a fallacy of composition to use that inherently finite point to bind the set as a whole when its essence is endlessness.

    I don’t think there’s any fallacy of composition going on here. Every natural number is finite, but no one is saying that N is in any way finite.

  196. DS, I think if you look carefully, you will see that I am saying something a bit more precise about just what an induction that chains from a finite initial case C_0 or C_1 etc means. And notice the issue just put up of limit ordinals with w the first precisely as it is successor to endlessness. KF

  197. Aleta, I believe per fair comment that my answer was given by way of pondering what is involved in induction from the sort of initial finite case C-0 or C-1 means. Unlimited succession, which brings back in the very issue at stake. Okay, I have spent much time here, the evolving local situation calls for my attention as the clock ticks on. Please note my fishing for what are the naturals. KF

  198. “Please note my fishing for what are the naturals. KF”

    What does that mean?

  199. KF,

    DS, I think if you look carefully, you will see that I am saying something a bit more precise about just what an induction that chains from a finite initial case C_0 or C_1 etc means.

    Well, do you accept that the standard inductive proof given by Tao does indeed show that all natural numbers are finite? Your phrasing above suggests not.

    And notice the issue just put up of limit ordinals with w the first precisely as it is successor to endlessness. KF

    I have mentioned limit ordinals/cardinals several times already, so it’s not exactly new to the conversation. I would also be cautious of calling a limit ordinal the “successor” to anything, either “endlessness” or “an endless set”, since it by definition is not a successor ordinal.

  200. Aleta, passed by for a moment, cf 195 above. Bolded, in context. KF

  201. kf, you write,

    The naturals are those counting set numbers in principle reachable by stepwise sucession from {} –> 0, {0} –> 1, {0,1} –> 2 etc, noting that they are endless in succession. And here I am using definition in a conceptual, even philosophical sense, fishing for the inherent, apt description of the essential meaning of a phenomenon, i.e. I am feeling for the innate nature of these numbers not imposing an arbitrary axiom that then controls what the meaning is. That endlessness in the definition — amplified from the ellipsis — implies that no stepwise process can exhaust them

    I agree with the non-bolded parts above. However, the part I bolded doesn’t make sense to me. What “arbitrary axiom” are you referring to? The definition of the naturals as being created by the statement that every natural number k has a successor k + 1 is an axiom, but it is not an arbitrary axiom – it is a universally accepted, I think, part of the foundational definition of natural numbers.

    But when you write that you are “fishing for the inherent, apt description of the essential meaning of a phenomenon, i.e. I am feeling for the innate nature of these number …”, I think you are talking about a psychological issue, not a mathematical one. The mathematical issue is clear, as stated (mostly) in the non-bolded part above: the psychological issue – what I referred to in an early post as the mystery of the infinite, is what you are grappling with and fishing for.

    I understand, I think, both as a math teacher and as someone who has tackled various mathematical problems myself, this issue of searching for a sense of “really understanding.” In teaching calculus, I have often seem students who learn to do the math correctly, but who don’t really grasp what it is about. The math itself is what it is, but a sense of comprehending what it is really all about is a feeling that for some students is always vague, and for others develops, sometimes suddenly like a light bulb.

  202. It would appear that the infinite must be simple and cannot be composed of parts or anything that can be counted.

  203. KF @193:

    And the continuum of [0,1] would then allow filling in the gap between w + g and W + (g + 1) by suitable extension of the number system model, however informally I have argued. That is, I can see the reals or hyper reals if you will filling in between the transfinite ordinals endlessly too; there is plenty of room at the bottom near 0 as between any two neighbouring values m and n down there, there will be another, say p.

    Wha?? We’re having a hard enough time dealing with N here. I suggest we keep our eyes on the ball and not indulge in any such fanciful constructions for the time being.

  204. DS, passed by a moment, a quick point. If a model being explored may have an interesting, possibly useful property it is worth noting. Catapulting and implications in this exploratory sandbox “space” are interesting — and they connect to, just how SHOULD we understand naturals, reals and the two infinities — big and small. Notice, endlesness is IN the naturals, that is how we get to w, and the real continuum between 0 and 1 opens up catapult phenomena that bridge to the transfinite once infinitesimals are on the table too. KF

  205. Aleta, I am looking at numbers that naturally appear, hence their name. From these we go to rationals, mixed numbers, place value/power series systems [think, decimals], reals, complex, then structures such as vectors, matrices etc. So, if they are natural, what is that nature? As opposed to one sets up arbitrary clusters of axioms and explores in a sort of super crossword puzzle game. Pardon, still mainly attending to local affairs — and, e.g. the spin games to make pseudo consultations sound like the real deal are “interesting.” KF

  206. PS: Just for now, I toss in, how do we get to endlessness if the nature of the finite is to be ended?

  207. kf, I am absolutely sure you are aware that we having been using “natural number” in the mathematical sense, not in some informal sense as including all sorts of different types of other numbers. As Dave said, let’s keep our eye on the ball: N = {1, 2, 3, …} is the topic.

  208. Aleta, I note, the issue of what the naturals properly are is pivotal to their extent and to the issue of ending the endless, up-ways or down-ways. As for the focus of the thread as a whole, this is just one facet. The OP sets the focus, in the end. KF

  209. KF,

    DS, passed by a moment, a quick point. If a model being explored may have an interesting, possibly useful property it is worth noting. Catapulting and implications in this exploratory sandbox “space” are interesting — and they connect to, just how SHOULD we understand naturals, reals and the two infinities — big and small. Notice, endlesness is IN the naturals, that is how we get to w, and the real continuum between 0 and 1 opens up catapult phenomena that bridge to the transfinite once infinitesimals are on the table too. KF

    Exploration is fine, but I do find it somewhat odd that while you have serious doubts about something so straightforward as mathematical induction, you nevertheless seem quite confident in this “catapult” idea, which is not exactly rigorous, to say the least.

    Your idea of filling in between ordinals can be made rigorous, I think (see the long line for something similar) but I don’t know how it’s going to shed any light on N.

    PS: Just for now, I toss in, how do we get to endlessness if the nature of the finite is to be ended?

    How can mirrors be real if our eyes aren’t real? j/k.

    Seriously, though, I have no idea what this question is asking. At least if we stick to N, we can rely on standard definitions that, hopefully, we all agree on.

  210. The OP was about “step by step causal succession” and “counting to infinity.” Step-by-step and counting are done with the natural numbers (or the integers if you count backwards.) The focus of the OP is the natural numbers, not other types of numbers.

    So I ask you to address this simple argument directly, rather then just saying “it’s not so simple”.

    if w (or the corresponding aleph null) are the first transfinite numbers, and they are not in N, then that proves that all natural numbers are finite. Q.E.D.

    and, changing the wording to be declarative,

    If all transfinite numbers (cardinal and ordinal) are not actually members of N, then it follows that all numbers in N are finite. Either all natural numbers are finite, or there are transfinite numbers in N. If we have eliminated the second possibility (which you agree we have), the first statement is true.

    Why is this argument not valid?

  211. N = {1, 2, 3, …} is the topic.

    Reminds me of this book:

    One Two Three . . . Infinity: Facts and Speculations of Science

  212. Wow – I had forgotten about that book, but it was one of the first to get me really interested in math. I also really liked Isaac Asimov’s books on science – I still have a whole set of his books. These were formative in my early teen years.

  213. DS, the same “catapult” is used routinely in non standard analysis, as has been pointed out to you already. Used, to bridge the all but zero and the transfinitely large. In particular, infinitesimals and their multiplicative inverses the hyper reals. This last includes hyper integers. Hyper reals have to be continuous and the continuity of [0,1] makes for that when I looked at what I have called mild infinitesimals. This BTW fits in with the use of that interval in looking at the continuum and its degree of endlessness as distinct from that of counting numbers. Calculus and infinitesimals, I first saw in 4th form. Seeing the Newton-Leibniz approach reborn through non standard analysis opens up vistas. There is good reason to be confident that the use of 1/x ad multiplicative inverses bridges the v large and the v small. I am just suggesting what if a mild infinitesimal m through 1/m = A, gets you to w + g, g a large but finite number that is a successor to w, w + 1 . . . w + g . . . EoE . . . KF

  214. zxc
    _____
    Aleta & DS:

    Aleta, 196: >>back to 192: if all known transfinite numbers (cardinal and ordinal) are not actually members of N, then does it not follow that all numbers in N are finite? What alternaives are there? Either all natural numbers are finite, or there are transfinite numbers in N. If we have eliminated the second possibility, the first statement is true.

    “Jarring” concerns about an endless number of finite numbers may express a natural (which is sort of a double entendre, I guess) sense of the mystery of infinity, one which Cantor et al plumbed for us. But unless you can actually establish something mathematical (which my post at 192 highlights), your sense of being jarred is waiting for you to come to terms with that mystery.>>

    I long ago learned to respect my sense of cognitive dissonance, of logical incongruity. Remember, as primarily a physicist I had to swallow the transition from the classical to quantum and relativity.

    In this case, it is not that logical chaining from initial values and an assertion C-n for case n that then entails C-n+1 that is a problem. Yes, it chains on, unlimited. But it cannot exhaust, it is stepwise and subject to the ellipsis of endlessness.

    With what is on the table here being the very counting set that serves as first yardstick of endlessness in the first degree, Aleph null.

    The claims being made come too close to ending the endless.

    I elaborate a bit more in . . .

    DS, 197: >>It seems to me if you reject the inductive proof that all natural numbers are finite, then you reject the validity of mathematical induction in general. And since N itself is generally defined inductively/recursively, I don’t see how you make sense of the existence of N as a set itself . . . . Every natural number is finite, but no one is saying that N is in any way finite.>>

    I took a careful look at what the usual proof by induction I have used since 6th form days actually establishes.

    It finds a case 0 or 1, then establishes — on a framework for case-n (C-n) that C-k => C-k+1. This hangs a logical chain on the first fact and proceeds to do so STEPWISE. This establishes unlimited extension indeed but again we cannot end the endless through a chaining of successive finite steps. We can only append a pointing ellipsis of endlessness.

    The issue is thus endlessness as the heart of the transfinite nature of the successive counting numbers.

    Which can be pointed to but not exhausted.

    Now, look at the set {0, 1, 2, . . . EoE . . . }

    This is, endless. It is also the sequence of counting sets per the von Neumann type assignment

    {} –> 0, {0} –> 1, {0,1} –> 2 . . . EoE . . .

    such that any value such as 5 emerges as the order type of its predecessors collected:

    {0,1,2,3,4} –> 5

    This means that the counting, 1:1 match etc properties of the successor and its cardinality are established by what is on the LHS of the assignment. It can be shown that this example is WLOG.

    Continue to endlessness and w emerges as supremum:

    {0, 1, 2, . . . EoE . . . } –> w

    The endlessness is in the LHS, the set defined as the natural numbers.

    Endlessness is the heart of being transfinite and we need to face it. Where, our process of counting here must be endless on the LHS.

    By definition, ordinary mathematical induction points to but cannot exhaust endlessness.

    The claim that all naturals are finite hangs from finitude of the first and an endless succession that can only be pointed to.

    But in looking at k and k+1 for instance, the endless chain begins again any number of times and points to onward endlessness that can be put in 1:1 correspondence with the set starting from 0, 1 etc. The set is transfinite, on the LHS.

    Does finitude then chain to and exhaust the members?

    By mere force of instant logical extension to the whole?

    Thus entailing, all counting sets in the succession are finite, never mind the endless chain of such counting sets that scale ever upwards in succession, a process which must — at least, it seems it must — in the far zone go on to endlessness in the sets?

    [Or, the successive count has not become transfinite, so far as I can see to date. How do we get a transfinite collection of ever-mounting distinct counting sets where each and every one is finite, including at the far zone of endlessness? To my mind so far, I can see that we have a stepwise incremental algorithm of chaining, that is in an endless loop that it cannot exhaust. It spits out a step per loop, and points on to endlessness but is in itself inevitably finite, it can only step forward a finite, countable number of times. It points to potential infinity, it does not actually exhaust it. It also establishes that at each step we have the next successive number as label for the set that collects sets thus far, and only if the counting sets in the far zone scale to the transfinite within their internal membership lists can we have an overall set that is endless and transfinite. In short, there is nesting of the emerging transfinite character of the whole. At least, that is what I am seeing.]

    The problem is, ending the endless.

    Finitude implies ending, chaining in steps is inherently finite but points to the endless succession. And w is the limit ordinal that summarises that endlessness and is successor to the endless chain.

    There is no finite k such that k + 1 = w.

    Instead, I suggest a more modest interpretation: induction shows an unlimited, reliable logical chain that will work for any particular n you please, which by being specified becomes inevitably finite.

    But it cannot exhaust the endless.

    Just as w cannot simply succeed any particular k.

    Just as, the successive counting set looping algorithm can only ever stepwise attain to the finite and in so doing ever extends the scale of the sets, but points to the endless succession.

    The claim that finitude spreads through the whole chain by succession seems to me to suggest a claim to exhaust, or end the endless by an algorithm that can only ever be actually finite.

    For the moment, I think it safer to say, that finitude propagates down the chain, without limit as our algorithm loop counter increments and labels progress since case 0 so far. But that is very different from claiming it can exhaust the set as a whole.

    If it did so, it would indeed confer finitude upon all members, but to do that it has to end the endless, which by its very nature it cannot, it ever invites another clock-tick and step.

    Further to this, we are counting using this particular set [here, I envision the loop counter and a printer spitting out the assignment endlessly on Mr Turing’s paper strip or a modern update thereto], and a finite number has the character, it is ended and surpassed even. If every counting number in succession is finite, how then can we consistently claim the set as a whole is transfinite, endless?

    So, I put my qualms on the table, I asterisk the claim.

    I do not dismiss it as absurd — but I am concerned as to its coherence given the inherent issue that the counting numbers in succession are inherently endless.

    I do not know if we are able to bridge to mutual understanding as to why my concern. But maybe the idealised processor chugging away endlessly at a counting and printing loop algorithm but utterly unable to exhaust the set as a whole, stamping the final member as finite — whoops there goes another clock tick — can help.

    Where, I am confident that such is a reasonable mathematical exercise of induction from the particular to a succession rule to the chaining to unlimited extent. Howbeit, the chaining inherently cannot exhaust the whole. And that whole must be transfinite and to do so it looks a lot like the far zone we cannot reach has to have a fractal-like, nested copy of the whole in it.

    And, w is not somehow of distinct quality and characteristics from what lies on the LHS of the assignment of identity and labelling as ordinal successor that sums up and holds the cardinality of what it now tags:

    {0, 1, 2, . . . EoE . . . } –> w

    DS, 201: >>do you accept that the standard inductive proof given by Tao does indeed show that all natural numbers are finite? Your phrasing above suggests not.>>

    Please see the thought exercise of a counting loop algorithm implementing machine to see how I think unlimited reliable succession is not equal to ending the endless.

    Notice the need to embed a copy of the whole to end the process.

    Potential, but not actualised.

    Pointing to, but not completing, in a context where completing is the requisite.

    In short, I fear we are overclaiming what induction per se delivers.

    Aleta, 203: >>The definition of the naturals as being created by the statement that every natural number k has a successor k + 1 is an axiom, but it is not an arbitrary axiom – it is a universally accepted, I think, part of the foundational definition of natural numbers. But when you write that you are “fishing for the inherent, apt description of the essential meaning of a phenomenon, i.e. I am feeling for the innate nature of these number …”, I think you are talking about a psychological issue, not a mathematical one.>>

    Not psychological, but conceptual, philosophical, logical and mathematical by virtue of that discipline being the logical, abstract study of structure and quantity.

    Coherence is the chief test and guardian in such an exercise, but in this case there must also be congruence with the natural experiential root of whole counting numbers.

    Here, the issue seems to in part turn on unlimited though at any achieved stage finite succession of incrementing counting sets and ending the endless.

    Which by extension extends to causal succession from a claimed unlimited extension of antecedents [algorithmic process bridges readily to machine implementation] and to attempted decrementing from the transfinite just as much as incrementing to attempt attainment of the transfinite.

    Aleta, 209: >> we having been using “natural number” in the mathematical sense, not in some informal sense as including all sorts of different types of other numbers. As Dave said, let’s keep our eye on the ball: N = {1, 2, 3, …} is the topic.>>

    The mathematical sense of natural numbers builds on and must not violate what has been established through human experience of the phenomenon of matched counting sets and unmatched ones leading to counting, the concept of counting numbers that label particular standard sets — which are then made abstract — and so forth.

    The formal builds on and systematises the informal.

    It then leads to extensions: rationals, reals, complex numbers etc. So, onward links are relevant. Especially given the context set in the OP.

    DS, 211: >>How can mirrors be real if our eyes aren’t real?>>

    Mirrors and eyes are both real.

    And my first look into an abstract, virtual half-infinite in principle world was when I looked in a mirror and learned that images can be physically located behind it by parallax.

    P* —> |: + + + > * Im (same distance behind, on line of norm)

    I recall once setting as a 6th form exercise, doing the pins and mirrors expt they did in 4th form, then challenging to ponder the virtual half-universe.

    Next step, set up two mirrors in parallel: endless, receding mutual reflections, in an endless in principle loop of light.

    In praxis, fading off as the reflections are not perfect.

    Applicable to laser cavities and creation of coherent radiation — and the place of half silvered mirrors leading to spiking thence q-switching by various means to get controlled, much stronger pulses.

    Mathematical, idealised extension: endless loops pointing to the infinite.

    I recall, looking into the recession as the two little mirror strips were put in parallel was quite a shock that opened up the vista of the infinite. Even, though it could not actually attain it physically.

    A crucial distinction between physics and mathematics.

    Which brings us back to our ever looping incrementing algorithm, which logically is unlimited but is in principle strictly forbidden from claiming to have ended the endless.

    Aleta, 212: >>Why is this argument not valid?>>

    Please see the algorithm loop based illustration of the distinction between unlimited extendability and ending the endless. I am concerned that we may have gone a bridge too far.

    KF

    PS: White screen of captcha death, I have to go back and do the copy, cut out, nonsense phrase and insert real post on edit trick.

  215. PS: Let’s represent that algorithm and logic machine:

    START –>

    1] Initiating Feed:

    Initial condition: {} –> 0 ===> LOGIC MACHINE, LM

    Initialise LM space for storing current counting set list { . . . }, here, initially to empty set then on increment to immediately following successors to go through 1, 2, . . . EoE . . .

    Initialise LM space for storing current assigned numeral for current counting set, here, the empty set

    Initialise printer, confirm ready

    Go on to fetch, decode, execute . . .

    2] LM-0:

    Set LM counter –> 0

    Print “{“, print list from counting sequence to date, comma separated values, print “} –>”, print [counter contents]// gives counting set assignment and states the successor

    Increment printer output sheet for next line.

    Go on to fetch, decode, execute . . .

    3] LM-next case:

    Increment LM counter value using standard, place value notation as stored in the machine

    Extend LM space for storing current counting set list { . . . }, to include newly incremented counter value // extends the counting set for the onward successor

    Print “{“, print just extended list from counting sequence to date, comma separated values, print “} –>”, print [current counter contents] // prints the result with the new counting set, preparatory to the onward successor

    Increment printer output sheet for next line.

    Go on to fetch, decode, execute . . .

    4] Continue:

    Go on to fetch, decode, execute code block 3 just above.

    This process is unlimited but inevitably is finite and however rapidly executed CANNOT exhaust and end the endless.

    Step 4 guarantees that, by imposing an endless loop.

    Endless loops are great for the main machine process in a computer, but these by definition are open ended and incomplete up to imposing a close down interrupt by shutting down the system or yanking the plug etc.

    And, this is in effect a proof by induction on initial case plus chaining logic plus standard result process at each step, here an algorithm rather than churning out the result of a formula etc.

    Which, we must recognise as unlimited but not ending the endless.

    And as an internal loop were it to do so the “final” printed set would be the whole endless set, nested.

    We RELIABLY produce instead an unending chain of finite counting sets but we do not have ability to exhaust the set as a whole.

    We revert to ellipsis of endlessness and collect the whole in a set we term the natural numbers big-N, giving its successor the finite symbolic numeral omega, then we start over again at transfinite level and proceed.

    Indeed the above exercise (it is patently code-able, save for the implications for storage) can be modified to execute that and to terminate at a suitable point after listing the ordinals in the compressed way say Wolfram did.

    That is a terminating, finite, exhaustively computable process that exploits the ellipsis of endlessness.

    BTW, thus the place for catapults that are not step by step incrementing processes. Using 1/x, 1 –> 1, 1/10 –> 10 [two orders of mag . . think in terms of place value steps], 1/100 –> 100 [4 orders], 10^-300 –> 10^300, [600 orders of mag], and then go on to the infinitesimal zone and catapult to the transfinite.

    Catapulting like that gives me a structural, quantitative connexion — which seems logically coherent — so that I can see the links from a bound and limited zone, [0,1] that is well within the finite set of ordinals and the transfinite that is beyond stepwise reach though steps point there.

  216. KF,

    DS, the same “catapult” is used routinely in non standard analysis, as has been pointed out to you already.

    What I’m saying is that:

    1/m = A, gets you to w + g

    is an impossibility. It’s like saying 2 + 2 = 5. The infinite hyperreals are not ordinal numbers.

    I would urge that you exercise the same caution here with the “catapult” that you do when discussing mathematical induction.

  217. KF,

    Continue to endlessness and w emerges as supremum:

    {0, 1, 2, . . . EoE . . . } –> w

    In fact, {0, 1, 2, …} = ω is how this is usually expressed.

    I will ask again: If you have problems with an induction proof actually “finishing”, what sense does it make to you to think of N as being a “completed” set in the first place? The set itself is generated by induction, after all. And no Turing machine can generate and list all the members of N in a finite number of steps.

    I think a more consistent position for you would be to deny the existence of N (in the sense that Aleta and I think of it) and just say one can work with finite subsets of N only. That would sidestep this issue of mathematical induction we are discussing.

    This is quite an unorthodox position, of course, but you wouldn’t be the only adherent.

    Edit: Further to my #218: One way to see a difference between ordinals and hyperintegers is this: There is no smallest infinite hyperinteger, while ω is the smallest infinite ordinal.

  218. DS, other numbers are drawn out of the ordinals. And as discussed I spoke to a mild infinitesimal taken through 1/m to get to A, which per an exploratory model I substituted as w + g. KF

  219. KF,

    In your notation, m must be a non-real hyperreal infinitesimal. I don’t have any idea how these things are graded as “mild” or not, so I will set that aside.

    In that case, 1/m = A is an infinite hyperreal, and is thus not equal to ω + g for any nonnegative integer g.

  220. DS, I have no problem with endlessness of counting numbers, and I have no problem with inductions being of unlimited extent, such that for any concrete substitution case it will hold. The issue at first level is completing the endless in that way. In that context w emerges as the assigned successor to the endless succession. And the issue is just what is the set of counting numbers starting from zero, and what are their attributes. The outlined stepwise process above distinguishes between being unlimited in successive extension and completing the set, nesting that to complete one would have to produce an endless copy of the set, which by definition is infinite and endless. Cannot be done. Analytically. Also, the endlessness is within the set itself, w does not add endlessness to it. Further as the way the set progresses is through collecting more and more successors, it will in process be finite but points beyond to endlessness. That whole context leaves me very wary when an inherently finite and potentially infinite only proof process is held to entail an endless actual number of finite numbers. There is a contradiction there or else something too close to be comfortable. KF

  221. Hi kf. I appreciate that you are genuinely working to communicate some perhaps ineffable issues concerning the infinite nature of the natural numbers.

    You write,

    I long ago learned to respect my sense of cognitive dissonance, of logical incongruity. Remember, as primarily a physicist I had to swallow the transition from the classical to quantum and relativity.

    Recognizing cognitive dissonance is important, because it is better to acknowledge two competing views than it is to deny one in order to relieve the uncertainty and conflict.

    However, as has been the experience for many trying to grasp some of the results of quantum physics and relativity, sometimes the truth is that one has to accept both perspectives in order to really understand the larger picture.

    I think the same is true of the infinity of the natural numbers. Each natural number is finite and there is an infinite number of them seems, perhaps, to set up an either/or cognitive conflict. You, I think, are trying to resolve the conflict by somehow denying one half of the and statement about rather then transcending the conflict and accepting the bigger picture. Like many Gestalt-ish issues, focusing too hard on one half of the picture makes it impossible to see the other half, but standing back and relaxing the vision allows one to see that there is a whole that encompasses the two different perspectives.

    With that said, throughout your long post you return often to the issue that I think is most bothering you. From multiple places:

    The claims being made come too close to ending the endless.

    The claim that finitude spreads through the whole chain by succession seems to me to suggest a claim to exhaust, or end the endless by an algorithm that can only ever be actually finite.

    Induction shows an unlimited, reliable logical chain that will work for any particular n you please, which by being specified becomes inevitably finite. But it cannot exhaust the endless.

    If it did so, it would indeed confer finitude upon all members, but to do that it has to end the endless.

    If every counting number in succession is finite, how then can we consistently claim the set as a whole is transfinite, endless?

    Pointing to, but not completing, in a context where completing is the requisite.

    This process is unlimited but inevitably is finite and however rapidly executed CANNOT exhaust and end the endless.

    Those are just some of the lines that you wrote that summarize your concern: that somehow claiming that every natural number is finite brings an end that contradicts the endlessness – the infinitude – of the set.

    I don’t believe this is a contradiction. I cant say anything more than I’ve already said to relieve you of you concern other than I think you thinking there is an issue here when in fact there is isn’t.

    No, you can’t exhaust or end the endless, but claiming that each step is finite doesn’t imply that you can.

    I think it is good and constructive that the discussion has finally, in my opinion, clearly delineated what the issue is, and where we have a difference.

    I see no conflict or contradiction where you do.

    I really think this is a Gestalt issue. Like the famous picture of the two faces/vase, when you see the finite nature of each step, you can’t see the endlessness, and when you think about the endlessness, you can’t see how all the components can be finite.

    Infinity is not a topic that our minds can natively comprehend, any more that is wave/particle duality, or non-simultaneity. But that is the beauty of mathematics – we can logically create systems that bring us to understandings that go beyond what we can intuitively grasp. Cantor et al, in formalizing notions of infinity, made us accept many ideas that are counter-intuitive, or in conflict with other notions that seem clear to us. But accept them we must, because the math shows us that they are as they are.

    So asserting that every natural number is finite, and that the set of all such numbers is infinite, does not claim, or imply, that we have brought an end to endlessness. Rather, it tells us something about the nature of endlessness.

  222. DS, If I am going to accept a paradox that runs brushingly close to a contradiction, you had better believe I am going to look at it very closely indeed to ensure that what I acknowledge is not in fact incoherent. That is for instance why you will sometimes hear me speaking of the deep nature of the problem of the one and the many. In theology the triune concept of God is another similar case. KF

  223. Aleta, the issue of a transfinite number of finites [successively arrived at — cf my use of a thought exercise logic machine earlier today], esp where counting sets extend without limit is right at the heart of my concern. Unlimited extension is one thing, ending the endless is another. KF

  224. DS, my problem with suggesting non real infinitesimal hyper reals — other than as a model building artifice to create a system, then be checked to see if the scaffolding can thereafter be removed — is that [0,1] is a continuum of reals. 0 is a real, and numbers in its very near neighbourhood should also be reals, per continuum. So, on the face of it, there is plenty room at the bottom near 0 to catapult infinitesimals that are all but zero. And of course, similarly proposing hyper reals beyond the reals should fit in with this. KF

    PS: Let me clip a discussion that brings out points where concerns pop up:

    http://2000clicks.com/mathhelp.....rreal.aspx

    Any of a colossal set of numbers, also known as nonstandard reals, that includes not only all the real numbers but also certain classes of infinitely large (see infinity) and infinitesimal numbers as well. Hyperreals emerged in the 1960s from the work of Abraham Robinson who showed how infinitely large and infinitesimal numbers can be rigorously defined and developed in what is called nonstandard analysis. Because hyperreals represent an extension of the real numbers, R, they are usually denoted by *R.

    Hyperreals include all the reals (in the technical sense that they form an ordered field containing the reals as a subfield) and they also contain infinitely many other numbers that are either infinitely large (numbers whose absolute value is greater than any positive real number) or infinitely small (numbers whose absolute value is less than any positive real number). No infinitely large number exists in the real number system and the only real infinitesimal is zero. But in the hyperreal system, it turns out that that each real number is surrounded by a cloud of hyperreals that are infinitely close to it; the cloud around zero consists of the infinitesimals themselves. Conversely, every (finite) hyperreal number x is infinitely close to exactly one real number, which is called its standard part, st(x). In other words, there exists one and only one real number st(x) such that x – st(x) is infinitesimal

  225. But no one is claiming to end the endless. Why you think that is the case is what I don’t understand.

  226. KF,

    DS, If I am going to accept a paradox that runs brushingly close to a contradiction, you had better believe I am going to look at it very closely indeed to ensure that what I acknowledge is not in fact incoherent.

    Could you state in precise mathematical terms what this alleged “paradox” is?

    DS, my problem with suggesting non real infinitesimal hyper reals — other than as a model building artifice to create a system, then be checked to see if the scaffolding can thereafter be removed — is that [0,1] is a continuum of reals. 0 is a real, and numbers in its very near neighbourhood should also be reals, per continuum. So, on the face of it, there is plenty room at the bottom near 0 to catapult infinitesimals that are all but zero. And of course, similarly proposing hyper reals beyond the reals should fit in with this. KF

    You can carry out any construction you want, as long as you do it correctly. I’m just saying that 1/m = ω + g is impossible under any scheme you have suggested so far, so you’ll have to try something else.

    I think all the info in the quote you posted concerning the hyperreals is already on the table, btw.

  227. F/N: Some reading going back to and upgrading Euler: http://www.maa.org/sites/defau.....p0075s.pdf KF

  228. Aleta, it seems to me that in a relevant context there are those who are explicitly claiming actual completion of an infinite stepwise succession. That context is surrounded by cases where issues and assertions may imply just such ending the endless. In those contexts issues on the meaning of the transfinite, ordinal succession to that zone, the nature of mathematical induction and of the set {0, 1, 2 . . . EoE* . . . } arises, joined to the onward reals, continuum and the interval [0,1]. In context we then see the transfinite ordinals from w and what may be connected therewith. KF

    *PS: I speak explicitly of ellipsis of endlessness as this seems critical. Notice, the ordinary form of mathematical induction uses step by step sequencing hung upon an initial value in a sequence of steps. Such a sequence extends without limit, but inherently cannot exhaust or end the endless.

  229. DS, My first concern is centred on the assertion rooted in an ordinary inductive proof/argument that natural, counting numbers [to which we may assign the succession of ordered counting sets starting from {} as discussed above] form an endless succession of finite values which are all finite but cumulatively belong to a transfinitely large set. KF

  230. Aleta, 223, thanks for the thought. I wish I did not see a point of concern much as you summarised by clipping. But, the concern is there — much like a theological doubt. Once there it has to be reasonably worked through. KF

  231. KF,

    DS, My first concern is centred on the assertion rooted in an ordinary inductive proof/argument that natural, counting numbers [to which we may assign the succession of ordered counting sets starting from {} as discussed above] form an endless succession of finite values which are all finite but cumulatively belong to a transfinitely large set. KF

    I just don’t see any problem with this. The individual finite natural numbers are very different from the entire collection.

  232. kf, you write,

    there are those who are explicitly claiming actual completion of an infinite stepwise succession.

    I’ve asked before: who is someone who is explicitly making this claim? Fundamental accepted mathematics does not make this claim, and I don’t know anyone who does.

    Is it possible that you are arguing against a position that in fact no one holds?

    Can you cite a source of someone ” claiming actual completion of an infinite stepwise succession”?

  233. George Gamow writes:

    The sequence of numbers (including the infinite ones!) now runs:

    1. 2. 3. 4. 5. …… &aleph;1 &aleph;2 &aleph;3 ……

    and we say “there are &aleph;1 points on a line” or “there are &aleph;2 different curves” …

    Is this a finite sequence?

    eta: weird. in the preview those came out as the Hebrew character א but not when saved.

  234. Here are some thoughts about the bigger picture in this discussion. At the end of this post, I’ll describe how I think this applies to our discussion about infinity.

    A major theme in the history of mathematics is that of extending the notion of number. I used to have an exploratory discussion day with my pre-calculus class about this. Here’s a brief summary, off the top of my head:

    1. Counting numbers starting with one, and rationals came first. There is pre-historical evidence of this understanding.

    2. Irrationals came next: the story of the proof that sqrt(2) is irrational is famous.

    3. Zero came next, introduced from the Hindu’s about 1000 AD, and introduced first as a placeholder in the decimal number system.

    Later, when the numbers came to be visualized on the number line, the counting numbers were backed down to zero rather than starting at one.

    Note that there was resistance to accepting zero as a number in this sense, because you can’t have zero things. Overcoming this resistance involved extending the concept of what numbers mean, moving them away from just associated with counting or measuring concrete objects.

    4. Interesting enough, negative numbers were next, and they didn’t get accepted as numbers in the Western world until the 1700’s: the argument being how can you have less than nothing?

    Here we see a pattern that will be repeated

    a. A number is impossible, often in respect to an equation: x + 5 = 3 has no solution because you can’t add a number that makes it less.

    b. Someone says lets invent a symbol for this “impossible” number, pretend that it exists, and see what happens: hence, –2.

    c. Mathematicians explore the possible rules for the new number, and it’s implications

    d. Mathematicians discover that there are no inconsistencies, and the new number fits well into the existing numbers system once understand how it works.

    e. Mathematicians find ways to both visually the new number and apply it to real-world situtaions

    f. The new number is fully integrated into mathematics, and our notions of number have been extended and have grown.

    5. Imaginary and complex numbers came next, not long after negatives, and the steps above were repeated. The equation x^2 = –1 has no solution, as a consequence of the rules for multiplying negative numbers, so let’s make up a number i = sqrt(-1) and see what happens.

    And, lo and behold, all sorts of stuff happens that works, fits i with the rest of the number system, and leads to all sorts of powerful applications and extremely counter-intuitive results such as the Mandelbrot set.

    6. So this brings us to infinity.

    The beginning idea is that of endlessness: a process that can always be continued. Building the natural numbers from each numbers successor is an example of an endless process. We say that, therefore, there are an infinite number of natural numbers. However, infinity isn’t a number at the end of the naturals, it isn’t a place to be reached, etc.

    However, Cantor decided to play the same game as above: let’s “pretend” that infinity is a number, let’s give it a name and symbol, let’s explore how and works, and see what we get.

    And again, we got new, consistent (for the most part) mathematics that introduced a new type of number, the transfinites. The transfinites extend the meaning of number. Just as negatives extended number past counting numbers, but did not change the nature of the naturals, and imaginary numbers extended the reals, but did not change them, the transfinites extended the concept of number to include infinity, but it did change or undo the basic nature of the other numbers.

    So aleph null is the name of the infinite number of natural numbers. That doesn’t change the fact that the naturals are defined by the successor rule, so that each and every natural is finite. Cantor’s extension of number to include aleph null and other transfinites doesn’t add a mystery to the naturals that wasn’t there before.

    Just because Cantor was able to successfully to invent new mathematics involving transfinites doesn’t mean that the infinite set of naturals has been or could be completed.

    kf continues to claim that “there are those who are explicitly claiming actual completion of an infinite stepwise succession,” and I have asked him to cite someone who believes this.

    Perhaps, and I offer this as a hypothesis, kf feels that way because he feels that the existence of aleph null implies a completion of the infinite. But it doesn’t.

    If this is not the source of kf’s claim/feelings, then his feelings come from elsewhere. If so, I again ask for an example of someone who claims the infinite can be completed.

  235. Aleta,

    Passing by a moment again.

    The survey on numbers is useful.

    I am surprised to still see questioning the issue of ending the endless as that has been a root issue behind several UD threads recently. As in, the proposal of an actually infinite past of the physical cosmos entails an actually completed infinite succession of causal stages that can be labelled in one way or another that translates pretty directly into completing the endless.

    You may not support that, and that has been acknowledged from the beginning when you said such.

    But the matter lurks.

    Scroll up to the OP, where Durston cites a case.

    In this or an earlier thread there was talk of infinite past oscillating universes.

    You may not be interested in the cosmology but it is material context and brings up the math on secondary issues.

    That math is of significance, and is worth discussing; but it is in fact incidental though connected to the logical and conceptual issues.

    It is not the existence of w and/or aleph null that are the issue, it is when things are affirmed or implied that point to stepwise traversal of the infinite. And that connects to concerns I have over how we think of induction. That is why in part I spent time looking at the logical Machine generating the stepwise succession. A result is, it goes on limitlessly, but still cannot traverse the transfinite in steps.

    Where, in looking at {0, 1, 2 . . . } –> w, the endlessness is there in the LHS. The RHS does not pop it out of thin air. That’s why I have stressed EoE. Succeeding k to k +1 does not span the endless, it points to the potential infinite, and indeed we can put k, k+1 etc in correspondence with the overall set, underscoring the endlessness.

    And I find myself further uncomfortable with the proposition that on an ordinary inductive proof it is shown that an endless set that counts up has in it only finite members. That runs very close to an outright statement of ending the endless.

    I find myself needing to look very closely at that and related matters.

    When I do so, I find further that a lot of scaffolding artifices are popping up surrounding infinitesimals, hyper reals, super reals and whatnot.

    Last I checked the reals are a continuum, and [0,1] is a real continuum, and yet infinitesimals are being discussed close to 0 as neighbours that are smaller than any real number.

    Orange flag at minimum.

    On another approach, the law of the excluded middle is being set aside.

    Red flag!

    Put in multiplicative inverses and I would see a catapult to the transfinite zone. But then the links between hyper or super reals and reals and established transfinites held to extend the counting numbers — which supposedly are a subset of the reals — look murky.

    Orange flag again.

    What about to taking some transfinite ordinal, say w + g and down-counting in stepwise succession? Again, things start to get delicate.

    Orange flag again.

    At minimum, there is caution, proceed with extreme caution.

    Okay, for now, exploratory modelling that tries out things to explore.

    Red-amber flags waving, we explore hoping to spot the quicksand patches before we tumble in.

    I think it is worth taking reals as continuous in [0,1] seriously and regarding infinitesimals as all but 0, not finitely different from 0. That cries out for multiplicative inverses that are transfinite and as I see a perfectly good sequence from w up, why not a mild one, m, that drops us to say w + g when catapulted through y = 1/x? Where would that take us for our purposes , , , in say the lines of thought explored by Euler? Of course, I called that A, long ago and saw that it would get us to a down count through w + (g -1) –> A~1, etc. As well continuity in [0,1] would by catapulting neighbours, allow filling in say w to w +1 etc, i.e. the exploratory, naive approach suggests that the transfinite ordinals can be looked at as mileposts on a transfinite continuum that extends from w on, with an EoE leading up from, 0, 1, 2 etc and conceptually traversed by w being successor to the counting set with the EoE. Interesting, though not a mathematical proof from first principles by lock-down steps. Next, attempting a downcount in steps to reach a finite neighbourhood of 0 would try to traverse the endless. A, A~1, etc will go in correspondence with the 0, 1, 2 etc and we are back to stepwise traversal of an ellipsis of endlessness.

    Maybe that lends some conceptual support to the idea of such a traversal running into the gap between unlimited succession and traversing the endless that the algorithm from this morning shows at was it 217.

    http://www.uncommondescent.com.....ent-597338

    Meanwhile, it looks like Euler was thinking in not very dissimilar but much more sophisticated ways that someone is trying to rehabilitate through hyper real thinking:

    http://www.maa.org/sites/defau.....p0075s.pdf

    So, let us see how we can connect some dots into a coherent whole, if that is possible.

    The firmest thing so far is it is futile to try to cumulatively traverse the endless in finite steps.

    KF

  236. KF,

    If I may address some of your points to Aleta,

    Last I checked the reals are a continuum, and [0,1] is a real continuum, and yet infinitesimals are being discussed close to 0 as neighbours that are smaller than any real number.

    Orange flag at minimum.

    What exactly is the orange flag here? As we’ve stated several times, there are no real infinitesimals, although they exist in the hyperreals.

    On another approach, the law of the excluded middle is being set aside.

    Red flag!

    ?? Where did this happen?

    What about to taking some transfinite ordinal, say w + g and down-counting in stepwise succession? Again, things start to get delicate.

    Orange flag again.

    Well, say you attempt to count down from ω. There’s nowhere to go, unless you skip almost all of N. Is that a problem somehow?

  237. DS, do you not see the problem that [0,1] is by all accounts a continuum, and putting up numbers next to 0 that are not reals? I can see an artifice of argument set up as scaffolding that says well infinitesimals are smaller than any real; but there is an apparent price being paid here: is [0,1] a continuum or not . . . and does this not mean that between two close neighbours that are distinct we can insert another point, basically by averaging the first place in the deep decimals where there is a difference or the like? Orange flag. Red flag is reserved for the alternative that talks in terms of sacrificing excluded middle. Maybe, working premise, an infinitesimal is smaller than any finitely large real, but not quite zero. That’s more or less the rule of thumb view I have seen used for decades, lurking behind the limit approach, which is obviously now the standard one. And it has historical roots. It seems some serious artifices had to be brought in to look at them again. KF

    PS: attempting the down-count beyond w is exactly my point, it hits the ellipsis of endlessness and cannot break the cardinality of first magnitude endlessness. Stepwise process is inherently finite and cannot traverse the EoE to reach a finite neighbourhood of 0.

    PPS: Look at smooth infinitesimal analysis for walkaway from LEM.

  238. KF,

    DS, do you not see the problem that [0,1] is by all accounts a continuum, and putting up numbers next to 0 that are not reals?

    No, I don’t think there is a problem with this construction.

    I can see an artifice of argument set up as scaffolding that says well infinitesimals are smaller than any real; but there is an apparent price being paid here: is [0,1] a continuum or not . . . and does this not mean that between two close neighbours that are distinct we can insert another point, basically by averaging the first place in the deep decimals where there is a difference or the like?

    No, that’s not what a continuum is [at least in reference to subsets of R]. Otherwise, the rational numbers would be a continuum, which they’re not.

    Red flag is reserved for the alternative that talks in terms of sacrificing excluded middle. Maybe, working premise, an infinitesimal is smaller than any finitely large real, but not quite zero. That’s more or less the rule of thumb view I have seen used for decades, lurking behind the limit approach, which is obviously now the standard one. And it has historical roots. It seems some serious artifices had to be brought in to look at them again. KF

    But where exactly is the law of the excluded middle being set aside? I’m just guessing here that you are saying something about numbers either being 0 or not 0? I don’t know.

    Edit:

    PS: attempting the down-count beyond w is exactly my point, it hits the ellipsis of endlessness and cannot break the cardinality of first magnitude endlessness. Stepwise process is inherently finite and cannot traverse the EoE to reach a finite neighbourhood of 0.

    That has been acknowledged all along, though. Counting down from ω to 0 is not involved in the Hilbert Hotel inspection tour.

  239. kf, you write,

    I am surprised to still see questioning the issue of ending the endless as that has been a root issue behind several UD threads recently. As in, the proposal of an actually infinite past of the physical cosmos entails an actually completed infinite succession of causal stages that can be labelled in one way or another that translates pretty directly into completing the endless.

    You may not support that, and that has been acknowledged from the beginning when you said such.

    But the matter lurks.

    I’m glad you acknowledge that I’ve not defended nor discussed the infinite past issue.

    But that is different than the issue of the naturals.

    The difference in the two situations is this: that in creating the naturals we build each number from its predecessor, endlessly, moving “upwards”, towards infinity.

    In the example of the past, one is claiming that somehow one could move up from negative infinity, which makes no sense.

    So, could you separate the assertion that all naturals are finite, and that the set of naturals is infinite, from the issue of an infinite past?

    And, if you separate the two issues, is there anyone claiming that there is a problem with “ending the endless” if we look at just the naturals, or is the only concern you have with the other issue concerning the past?

  240. DS, Maybe you do not have a problem, but I do. LNC problem: numbers next to 0 but below the reals where there is supposedly a continuum [thus no gaps or breaks etc], is inherently questionable and needs some reasonable resolution. So far, I see, artifice, a useful fiction that works around a ticklish situation. KF

  241. Aleta, form the very first I welcomed that you agreed with me on that one. However, I am not sure that that is a generally acknowledged point, given what we see Durston et al pointing to and as we saw Spitzer remark on earlier. And the discussion has to bear in mind that wider context. KF

  242. Seeing how you KF are still into physics I thought I should mention that the model I have finds two of possible four requirements for intelligence in the behavior of matter, with no need to “guess” (anymore?) and “confidence” that normally sets constants that would be the part consciously felt by what you can call God if you want. I have no solid evidence that the universe is this way, but from what I have for theory is possible. This illustration shows what I have, for modeling the behavior of matter as though it’s fine tuned by intelligence even though that is not necessarily the case. In either event it’s a novel scientific model to experiment with:

    https://sites.google.com/site/intelligenceprograms/Home/Causation.png

  243. KF,

    DS, Maybe you do not have a problem, but I do. LNC problem: numbers next to 0 but below the reals where there is supposedly a continuum [thus no gaps or breaks etc], is inherently questionable and needs some reasonable resolution.

    I think I see what you’re saying, but remember that the infinitesimals are “infinitely close” to 0, so it’s not as if they are sitting in some “gap” in the reals, which would contradict the definition of a continuum.

    Anyway, the hyperreals were constructed almost 70 years ago, so I think any serious outstanding issues would be resolved by now. Granted, it’s a strange set.

    But there certainly is no violation of the law of the excluded middle here.

  244. Yes I understand there is a wider context, but in the narrower context, what problem do you see with all natural numbers being finite, and the set of naturals having an infinite number of members? I understand there are issues with the broader context, but you have continually seemed to argue there are issues with the narrower context of the naturals.

    It seems to me it would be useful for you to clearly, more clearly than you have, explain in perhaps more precise mathematical language, what the issue with the naturals are.

    Issues with w, or an infinite past, or infinitesimals, are interesting, but bringing them up as a larger context doesn’t actually address the smaller one.

    If the smaller issue (the naturals) were actually being addressed, then the larger context might be interesting, but it looks to me like your continually returning to larger issues (w, infinite past, infinitesimals) is a way of avoiding the arguments about the naturals.

    Every time the discussion tries to narrow down on the naturals, you fall back on “but there are other issues.”

    So, I repeat,

    Could you separate the assertion that all naturals are finite, and that the set of naturals is infinite, from the issue of an infinite past?

    And, if you separate the two issues, is there anyone claiming that there is a problem with “ending the endless” if we look at just the naturals, or is the only concern you have with the other issue concerning the past?

  245. DS, it gets worse, do points “touch” — thus no gaps — and where is the continuum. The definition of continuum I learned way back was, effectively, between distinct neighbour points, you can always insert an intervening one, which strictly implies pico gaps at the bottom; oh boy. (one oddity, cynically suggest continuum is a myth and accept the infinitesimals as fitting the “final” gap when reduction goes to EoE, a monad or its kissing cousin.) Not nice. Smooth Infinitesimal analysis, at price of workarounds to the LEM, puts in in effect crudely pico segments of infinitesimal scale. They get us to at that scale curves are concatenations of straight segments. KF

  246. Aleta, the issue is perilous closeness to a contradiction between everybody finite — including the “far end” of the succession of counting sets, and the set is transfinite as the EoE shows. This is a way to try to put it. The connexion to timeline of cosmos is through the series so the math can be looked at on its own, implications lie where they fly. KF

  247. There is no “far end”. I give up! (But I’ve said that before, and then have come back, so we’ll see.)

    And, what does perilously close to a contradiction mean? What contradiction? Why can’t you be more precise. This is math. What is the contradiction?

  248. KF,

    DS, it gets worse, do points “touch” — thus no gaps — and where is the continuum.

    I don’t know what points “touching” would mean. I also believe that the hyperreals do form a linear continuum, just as R does.

    Smooth Infinitesimal analysis, at price of workarounds to the LEM, puts in in effect crudely pico segments of infinitesimal scale.

    If you’re talking about the “sloppy calculus” that is sometimes taught using infinitesimals, it is a bit of a cheat, but I suppose it’s useful in some situations.

    I was never taught calculus that way, so I don’t have much to say about it.

    On the other hand, the version based on nonstandard analysis is absolutely rigorous, with no logical problems.

  249. Aleta,

    forgive mis-stating, I used a spatial metaphor and meant to refer to the far “zone” where the ellipsis of endlessness would have to be traversed if a step by step process wee employed.

    Perilously close to contradiction is relative to the relationship between paradox and contradiction.

    Paradoxes routinely run close and may seem incongruous, but sometimes they do go over into contradiction, and one cannot be sure on an initial glance, or even after much close study.

    I have said it in many ways, that if all naturals are finite, that runs close to contradiction by way of ending the endless. For, the range of counting numbers as a whole is said to be endless and for that to be it seems “intuitively” that it should have in it members that reflect transfinite nature in themselves.

    Think, innate mile markers on a road built by a programed step by step machine [or maybe a machine capable of simultaneously creating the road from the origin to the far zone], here an endless linear one in a flat space.

    If the road is endless, will the markers ALL be at finite distances from the origin at say the famous Kingston Parish Church point of departure?

    If so, is that not inherently a limitation, an implicit finite terminus?

    Or, is there a way to say the roadbuilding machine runs out of steam but the road picks up at the far zone at “mile marker” w and so forth?

    If not, how can we have every marker at finitely remote distance AND at the same time, the road with the markers is endless?

    KF

  250. DS, Bell’s approach is not the rough and ready survival of the C17-18 approach that still sometimes surfaces, but another approach that addresses the issue of a point array and smoothness with nilpotent infinitesimals, etc. As one illustration, an infinitesimal is viewed as such that its square will be zero, an extension of the concept that was classically put as, the higher order terms are vanishingly small relative to first order infinitesimals, dx >> [dx]^2. Where obviously [10^-300]^2 = 10^-600, which is vanishingly small for most practical purposes relative to the original scale, though obviously this is a finite example; the point is the number of orders of mag down on unity will double on squaring. It is said to be just as rigorous as the other approaches, save that there is a workaround on LEM, which on what I gather would typically obtain WRT a phenomenon if there is some fuzziness or superposition in it so that distinct contrasts of W = {A | ~A} do not obtain. That is, there are now at least three significant schools of thought that provide alternative perspectives on Calculus foundations. KF

  251. kf, you write,

    For, the range of counting numbers as a whole is said to be endless and for that to be it seems “intuitively” that it should have in it members that reflect transfinite nature in themselves.

    You’re intuition is wrong, I think. You’ve already said that w, the first transfinite number, is not in N. Now you say your intuition says that there should be members of N that “reflect transfinite nature” (whatever that might mean.)

    That’s a contradiction right there. In math, proofs and precise steps of argument are needed – intuition doesn’t override proof.

    I’ve offered the following proof, and asked you why you don’t think it is valid

    1. Either all natural numbers are finite, or there are transfinite numbers in N.

    2. w (or the corresponding aleph null) is the first transfinite numbers: w is the ordinal successor of the natural numbers

    3. w is not in N

    4. Therefore, there are no transfinite numbers in N

    5. Therefore, all natural numbers are finite.

    Your intuition tells you there needs to be some kind of other number between finite and w, something that “reflects transfinite nature”, that is in N, but you can offer no specifics about what that might mean. If you could provide mathematical justification for this notion (as opposed to vague concerns about “ending the endless”), you would re-write mathematics.

    So here’s a question: Can you at least entertain the possibility that your intuition is wrong, and that Cantor et al are right?

  252. Aleta, please note the ellipsis of endlessness is on the LHS of the definition of w as successor. That is it is embedded in the set. That is where my concern is. W exists as summarising the order type of the endlessness, it is not popping that into existence out of thin air. If the set were not of transfinite span, w would not be of first order transfinite cardinality. And being or transfinite span with every mile post being finite is at minimum a concern. KF

    PS: The fact that I am not asserting contradiction but instead concern suffices to show I am aware that my concern may be wrong, but needs good reason to see why. And, in this general context Cantor was also wrong on key matters, hence the issue of naive set theory vs ZFC. (In short, no authority is better than his/her facts and reasoning backed up by underlying assumptions; I here reveal my Protestant heritage.)

  253. What is the concern? You keep saying that, but have no specifics. If every milepost isn’t finite, then what is the nature of a non-finite natural number?

    I ask the following:

    1. Address my proof in 253 above – how is it invalid?

    2. If every milepost isn’t finite, then what is the nature of a non-finite natural number? Be specific about that.

    3. Answer the bolded question. Can you even entertain the possibility that your intuitive concern is wrong, and that established mathematics is right? [I see that you later answered this question in a P.S., so you can ignore this.]

    [edited to more clearly state three questions.]

  254. KF,

    Thanks for the reference. And yes, I stand corrected, the wikipedia entry on smooth infinitesimal analysis does say it is based on logic without the law of the excluded middle.

    I wouldn’t characterize that as a “red flag”, however.

  255. DS, passing by a moment, the law of the excluded middle is one of the three pivotal principles connected to distinct identity that are rightly termed laws of thought or first principles of right reason. That’s why I will also have to fish around to see if there is not a distinct identity in some relevant sense. KF

  256. Aleta, the naturals span to endlessness. Yes, we conceive the endlessness whole and see this as expressing what we sum up as a first transfinite, but that character is in the set of whole counting numbers. As for continuum, I suggest the point is that for any arbitrary pair of close members of R there will always be more between. That will include say 1/pi in the relevant interval and in fact 1/pi^n where n is a whole number, all of which are not rationals. Rationals are reals but not all reals are rationals. KF

    PS: I spoke of LEM st aside for smooth infinitesimal analysis, cf Bell et al. That’s a third approach.

  257. KF,

    DS, passing by a moment, the law of the excluded middle is one of the three pivotal principles connected to distinct identity that are rightly termed laws of thought or first principles of right reason. That’s why I will also have to fish around to see if there is not a distinct identity in some relevant sense. KF

    Well, as you stated, this smooth infinitesimal analysis is regarded as just as rigorous as “standard” analysis.

    It appears this example shows that we need not restrict ourselves to classical logic only when doing mathematics. I assume there are many other such examples. [Edit: In fact I know there are, but I haven’t looked into this much.]

    If you can point out a real mathematical issue, that is, an instance where this approach yields incorrect results, then I’d like to hear it. Otherwise, I see no problems.

  258. kf writes,

    Aleta, the naturals span to endlessness. Yes, we conceive the endlessness whole and see this as expressing what we sum up as a first transfinite, but that character is in the set of whole counting numbers.

    I don’t believe this addresses either of my questions: what is wrong with my proof that all naturals are finite, and if you think otherwise, can you be specific about the nature of these non-finite natural numbers that are “past the ellipse.” Saying that set of the counting numbers has a transfinite character does not say anything specific about the particular numbers in the set. If they all aren’t finite, what are they?

    Also, I don’t know why you addressed the continuum – that doesn’t bear on the topic of the naturals.

  259. Aleta:

    Can you at least entertain the possibility that your intuition is wrong, and that Cantor et al are right?

    Cantor leads to logical inconsistencies. For example Cantor said that all countably infinite sets have the same cardinality, ie the same number of elements. Yet standard set subtraction proves that is not so:

    Let set A = {0,1,2,3,4,5,…}
    Let set B = {1,3,5,7,9,11,…}
    Let set C = {0.2.4.6.8.10,…}

    A – B = C, proving that all countably infinite sets do not have the same cardinality, ie the same number of elements. And only contrived mental gymnastics can get around that fact.

  260. Hi Virgil. Every other whole number is even. Does that mean there are twice as many whole numbers as evens? What do you think?

  261. Hi Aleta- First please respond to my post with something of substance and then I will get to your question.

  262. Two counters- one counts every second and the other counts every other second. The first counter will always have a higher count (double or close to it) than the second- always and forever.

  263. Virgil, your logic in 261 is wrong. You can’t use “standard set subtraction” on infinite sets. There are just as many positive evens {2, 4,6, …) as there are positive integers [1, 2, 3, …]. This is about as foundational of a universally accepted fact about infinite sets as there is.

    Sets A, B, and C in your example all have the same cardinality – aleph null, the first transfinite number, the first type of level or infinity.

  264. Aleta, You don’t have an argument. Just saying “You can’t use “standard set subtraction” on infinite sets”, is meaningless. You actually have to make a case and you cannot.

    ALSO- Two counters- one counts every second and the other counts every other second. The first counter will always have a higher count (double or close to it) than the second- always and forever. That is just another proof of my case.

  265. Virgil, I’m afraid you don’t know what you are talking about. Try reading here: https://en.wikipedia.org/wiki/Countable_set.

    I quote,

    For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that the number of even integers, which is the same as the number of odd integers, is also the same as the number of integers overall.

    That’s it for me with you.

  266. Aleta, You are just a blind parrot blowhard. I know what Cantor said. I also know that what I said refutes him. And I know that what you said about set subtraction of infinite sets is total BS.

    These represent two countably infinite sets:

    Two counters- one counts every second and the other counts every other second. The first counter will always have a higher count (double or close to it) than the second- always and forever.

    You lose.

  267. Why do my detractors think that just repeating what I am refuting refutes me? Talk about being mental midgets…

  268. Aleta:

    I have pointed out in 217 above, two days back, how an inductive proof simply postpones the point of making the ellipsis of endlessness, so that it is unlimited but does not span the endless.

    Such a proof is good enough for showing that for any particular n we please, some C(n) will be true as it rests on C(0) or C(1) and shows that C(k) => C(k+1), but an endless loop that advances in steps is still incapable of actually spanning the transfinite.

    Indeed, take some arbitrary k of very high value, then proceed to k+1 etc. Then, put in correspondence with the beginning of the sequence count C(0), C(1) etc. That is, we have a proof that after k we have not made any material progress towards the transfinite zone.

    Instead, we rely on the ellipsis of endlessness and say the potentially infinite transfers to the set of all n. That is an imposition, in fact what was shown was that for an unlimited range C(n) will hold, but not that it has spanned the transfinite.

    Normally, that is of no consequence, we in effect have an axiom — or a sub-axiom — imposed similar to the parallel lines one in classical Euclidean Geometry.

    However, in the particular case, we are dealing with the set of counting numbers itself.

    It is this set that we rely on to count and to take in all possible counts, and which the ellipsis of endlessness (note my repeated emphatic use of this full description and abbreviation EoE*) shows must continue endlessly.

    That endlessness is where the transfinite nature of its cardinality comes from.

    As already, repeatedly, pointed out, we then follow the pattern of the finite and assign a novel number w as the successor to the endless succession:

    {0,1,2 . . . EoE . . .} –> w

    does not pop transfinite-ness out of thin air, it is already present in the LHS, in the EoE. Indeed, that is what the successor operation repeatedly shows, e.g. (as was actually used above) we see how

    {0,1,2,3,4} –> 5

    by way of:

    {} –> 0
    {0} –> 1
    {0,1} –> 2
    etc.

    does not pop five-ness out of thin air on the RHS but labels a phenomenon inextricably present on the LHS.

    But, again, the set with the EoE is the set we use to count, and it is its successive members that create its span, right through the ellipsis. If all of its members are finite per the imposition of the axiom of completion of the EoE (and I know I am giving a novel, descriptive label), then we have a paradox at best, that a string of inherently finite incrementing counting sets is transfinite.

    But,

    if . . .

    p1: the string is created by finitely steps to have in it a finite value so far,

    then . . .

    c1: there is not endlessness.

    While, if . . .

    p2: the string has gone to endlessness,

    then . . .

    c2: the endless degree of steps must find itself reflected in the substance of the counting sets in it.

    where also . . .

    c3: the counting sets so far are all always collected on the LHS

    That is one thing that 217 showed, by showing the presence of a copy of the set so far at any given count in the LHS list of successive counting sets and their numeral representations.

    Allow me to copy that discussion, clipping 217:

    START –>

    1] Initiating Feed:

    Initial condition: {} –> 0 ===> LOGIC MACHINE, LM

    Initialise LM space for storing current counting set list { . . . }, here, initially to empty set then on increment to immediately following successors to go through 1, 2, . . . EoE . . .

    Initialise LM space for storing current assigned numeral for current counting set, here, the empty set

    Initialise printer, confirm ready

    Go on to fetch, decode, execute . . .

    2] LM-0:

    Set LM counter –> 0

    Print “{“, print list from counting sequence to date, comma separated values, print “} –>”, print [counter contents]// gives counting set assignment and states the successor

    Increment printer output sheet for next line.

    Go on to fetch, decode, execute . . .

    3] LM-next case:

    Increment LM counter value using standard, place value notation as stored in the machine

    Extend LM space for storing current counting set list { . . . }, to include newly incremented counter value // extends the counting set for the onward successor

    Print “{“, print just extended list from counting sequence to date, comma separated values, print “} –>”, print [current counter contents] // prints the result with the new counting set, preparatory to the onward successor

    Increment printer output sheet for next line.

    Go on to fetch, decode, execute . . .

    4] Continue:

    Go on to fetch, decode, execute code block 3 just above.

    This process is unlimited but inevitably is finite and however rapidly executed CANNOT exhaust and end the endless.

    Step 4 guarantees that, by imposing an endless loop.

    Endless loops are great for the main machine process in a computer, but these by definition are open ended and incomplete up to imposing a close down interrupt by shutting down the system or yanking the plug etc.

    And, this is in effect a proof by induction on initial case plus chaining logic plus standard result process at each step, here an algorithm rather than churning out the result of a formula etc.

    Which, we must recognise as unlimited but not ending the endless.

    And as an internal loop were it to do so the “final” printed set would be the whole endless set, nested.

    We RELIABLY produce instead an unending chain of finite counting sets but we do not have ability to exhaust the set as a whole.

    We revert to ellipsis of endlessness and collect the whole in a set we term the natural numbers big-N, giving its successor the finite symbolic numeral omega, then we start over again at transfinite level and proceed.

    Indeed the above exercise (it is patently code-able, save for the implications for storage) can be modified to execute that and to terminate at a suitable point after listing the ordinals in the compressed way say Wolfram did.

    That is a terminating, finite, exhaustively computable process that exploits the ellipsis of endlessness.

    As you can see, the axiom of the EoE is being used to decide the matter, carrying the whole weight of the conclusion that the counting sets are all finite but their span is transfinite. That is a big weight for a single — and often implicit — premise to carry.

    I suggest that what is shown is that there is an unlimited succession of incrementally lengthening counting sets which cannot be completed by a succession of actual successive steps, but we use a symbol, the EoE, to represent that there is a potentially infinite process here. Often we then have to address things like partial sums and error terms, showing that beyond a given point the onward difference to endlessness would be within a certain error band. That again uses EoE. It also implies that onward terms taper into infinitesimals trending to 0 such that the onward sum is within error bands. Or, in epsilon delta terms, the sequence of partial sums will be within some delta neighbourhood of the limit as pushed forward. Infinitesimals lurk, even when we find ways to avoid talking about them: he who must not be named and all of that.

    We need it, but we should be aware that we are tickling a dragon’s tail.

    On the case in view, we do not have a sequence of partial sums approaching a limit, we have a limitless expansion, indeed this ordered set gives us the metric for endlessness.

    For us to go to the RHS and assign w, the endlessness has to appear in successive subsets hidden under the EoE; especially as w is not the successor to any one given value, it is a recognised successor to the process as a whole.

    That is why I am uncomfortable with the idea of concluding that all subsets collected in succession are finite, to endless extent. A more direct conclusion is that to any degree we can complete the count succession, it will be finite, but the process is endless. And we posit a symbol and successor for that endless collection, w.

    When we term w the first transfinite ordinal and assign it as beyond the natural counting numbers, that is a definition that has to face the paradox above.

    We have to live with it, but should recognise it.

    Let me highlight, a summary of what I am comfortable with:

    Finite counting numbers extend without limit, we cannot count out and list the set that collects such numbers [i.e. it is a fallacy to assert or imply ending the endless in successive steps], the set as a whole can be symbolised by using EoE, and has cardinality first degree of endlessness. That first degree endlessness is symbolised as aleph null, and we collect the sets in succession as a whole from 0 and assign a fresh number that succeeds, w — which we explicitly recognise as transfinite.

    I think we should be cautious, then in using ordinary mathematical induction and the axiom of EoE to then say all naturals are finite and yet the set as a whole — composed by incremental succession such that the counting sets lurking under the symbols stretch out with the scale of the whole — is transfinite. At minimum, the implicit premise of EoE should be explained and the weight it carries should be disclosed.

    Of course, here is the axiom of infinity in its usual simple form:

    There is a set I that contains{} –> 0 as an element, and for each a (an element in I), the set a UNION {a} is also in I.

    That is, starting from {} –> 0, we collect in succession the preceding sets and extend endlessly. Such crucially depends on EoE to propagate to the full set.

    A set that as we see is inherently not completed by stepwise succession. So the [sub-]axiom of the ellipsis of endlessness is carrying a lot of weight.

    Which, we should duly note.

    And in turn that allows us to answer VC’s concern:

    N: 0, 1, 2, . . . n . . . EoE . . .

    Multiply each element by 2:

    E: 0, 2, 4, . . . 2n . . . EoE . . .

    That is this is a disguised form of the whole set thanks to EoE.

    Likewise, transform each n in N to 2n + 1:

    O: 1, 3, 5 . . . 2n + 1 . . . EoE . . .

    Again, a disguised form thanks to EoE.

    So, the three sets are in mutual 1:1 match.

    Likewise, shift k:

    N: 0, 1, 2 . . . EoE . . .

    s_K: k (= k + 0), k + 1, k + 2 . . . EoE . . .

    This is just a k-shifted transform of the same fundamental set. The symbols have changed but the sets are all countable and transfinite per EoE. They are fundamentally the same. There is self-similarity of sub-sets, a sort of fractal self-repeating in the smaller pattern in the loose sense.

    And so, when Cantor et al took the paradox and said, okay when a proper subset can be matched 1:1 with the set of counting numbers that is a way to recognise its transfinite nature, it opened up a new world.

    Of course, VC’s counters A and B with an underlying 1-second clock feed are extensions of the algorithm clipped from 217 above with one (say B) at a half-rate count, in effect there is an inner loop that sets up a 2-count then transfer to the counting register.

    With a common start B will run at more or less half the count of A to unlimited extent.

    Both are endless processes and show that a stepwise succession cannot complete endlessness. It also applies the [sub-]axiom of EoE and concludes that for any clock tick k, counter A will read k and counter B roughly or exactly half of k. But both are headed to the same transfinite zone. Just, B is the proverbial slow boat to China. Where at any given time, we will always be pointing onwards to a potential infinity.

    EoE is important.

    KF

    *PS: I think I should quietly note: your highlighting as if typical of an obvious typo in such a context above (pardon, I will sometimes make such errors and fail to spot/correct them), per fair comment, does not help the discussion. That is like the verbal slip of speaking of a far end rather than zone; also highlighted. Sorry for slips and typos, but they are no more than that. Just yesterday I found myself saying how decisions are *maken,* an inadvertent blend of made and taken. Oopsie, but worth a chuckle.

  269. F/N: all of this points to the issue of claimed or implied actual completion of the endless, which is where we started.

    Viewing the cosmos as causal succession:

    . . . C_k –> C_k+1 –> . . . C_n, now

    we see that a causal succession embeds a succession of states.

    Is the LHS ellipsis a completed EoE so that we are in the zone w + g i.e. the past was transfinite?

    Nope, as EoE cannot be bridged or traversed in finite steps. Language alone is already trying to warn us.

    A finitely remote initial point is indicated, as was discussed already.

    This is just a contextual reminder.

    Nor does it work to say at any p in the past we are only finitely remote onwards from k and we can repeat endlessly:

    . . . C_p –> C_p+1 –> . . . C_k –> C_k+1 –> . . . C_n, now

    No, the ellipsis on the LHS is still there and would still be endless.

    Yet worse is the case where one implies an endless causal succession in the past to the present, which if it means anything means that for some p’:

    . . . EoE . . . C_p’ –> C_p’+1 –> . . . EoE . . . C_k –> C_k+1 –> . . . C_n, now

    Ending the endless is a fallacy.

    If you doubt this, kindly show such an actual step by step completion or algorithm that can bridge the implied transfinite span in steps.

    We need to live with a world that manifests an inherently finite past succession to date.

    A world that strongly points to a beginning, where — let’s augment — a Root R gives rise to the beginning B, from which temporal-spatial causal succession proceeds:

    R:B –-> . . . C_k –> C_k+1 –> . . . C_n, now

    Where, further augmenting, R is a necessary being root.

    Just a refocus on context.

    KF

  270. KF,

    We need to live with a world that manifests an inherently finite past succession to date.

    However, assuming an infinite past, the manager of the Hilbert Hotel could, proceeding at a rate of one room per second, complete the inspection of every room today.

    Recall that this is a beginningless tour, not an endless tour.

  271. kf writes, “Just a refocus on context.”

    No, just a rehash of things already said multiple times, without addressing my main points in 253: a simple proof that all numbers in N are finite, which fails only if there is some third kind of number in N (not finite but not transfinite) that “reflects transfinite nature” and is somehow “beyond the ellipsis” yet still in N.

    However, kf doesn’t address the issue of trying to mathematically supply some specifics about these numbers, or any other part of what he thinks exists if all natural numbers aren’t finite.

    So, kf has a intuitive concern about something (“ending endlessness?”), but isn’t able to specifically or mathematically give it any coherence.

    I’ve tried pretty hard to understand his perspective, but that’s my summary of the situation. Since it doesn’t look like there will be any further progress, I believe (again) that this is the end of the discussion.

  272. DS, set up the algorithm and show it. KF

  273. Aleta, I have pointed out the concern and how it comes out given the implicit sub axiom of ellipsis of endlessness. Notice, I have shown that no step by step unlimited process can exhaust the endless [at any k, k+1 etc the process starts over again from effective start point and cannot span], so there is an implicit axiom that does the work. KF

  274. KF,

    Since this process is beginningless, it cannot be described by an algorithm.

    I have already given all relevant details of the manager’s tour: He was in room number -n, n seconds ago, for every natural number n. Clearly no rooms were missed, and the tour ends at the present.

    Note that there is no analog to the “ellipses of endlessness” in the HH. Edit: No ω’s either.

  275. kf, we agree: “No step by step unlimited process can exhaust the endless”, given that is what “endless” means. Going on endlessly doesn’t “span” anything – it just goes on and on and on …”

  276. DS, And so you are trying to span the endless in steps from the suggested infinite past. But, you do not even have a first step, just a claimed forever continuation. Where if infinite past means anything at all, it means that at some point in the causal succession there was an endless span of steps to be bridged to reach here. This is a contradiction in terms and in concept as well as a failure of the sequence to span in steps. If we cannot ascend to the endless and the transfinite, apply the mirror reflection, we cannot descend from it either. And in HGHI, the manager cannot inspect the rooms in toto in steps for the same reason. KF

  277. KF,

    I’m going to insist on no rephrasings here. If you have a critique, please use the same language I am using.

    What I’m saying is that the manager completes a tour of the HH presently, according to the schedule I laid out.

    The process is beginningless, it is true.

    Edit:

    Where if infinite past means anything at all, it means that at some point in the causal succession there was an endless span of steps to be bridged to reach here.

    No, that’s not what I understand it to mean. We’ve been over this repeatedly, but the manager was never more than finitely many rooms away from the front desk.

  278. KF,

    An infinite Turing Machine tape with a single end is a good model for the HH, with each cell representing a single room, and the “last” cell representing the front desk.

    Given such a tape, how many cells are infinitely many steps from the last cell? None, right?

    Edit: Here’s a picture. I am thinking of the tape oriented in the opposite direction so that the last cell is on the right (so the manager moves from left to right), but of course that makes no essential difference.

    Which cell is infinitely far from the end?

  279. DS, Let us look at two managers, one who starts at the far zone of the hotel infinity and inspects rooms, one from the near zone. Both set out at one room per second and have to inspect all rooms. Will either ever complete, why? KF

    PS: I add, an endless paper tape has rows of dots, as follows:

    r0, r1, r2 . . . . EoE . . .

    Whichever way you pass it through a read/write head — say it is the way clean/dirty rooms are recorded — it is the same, and to have been going from the infinite past and have reached a finitely remote k from 0 is like:

    . . . EoE . . . H:k+1, k, . . . 2, 1, 0.

    But to get to k, first you have to do:

    . . . EoE . . . H:p+1, p, . . . EoE . . . k+1, k, . . . 2, 1, 0

    You cannot traverse either of the two EoEs in steps.

    PPS: Any 0, k, or p will be infinitely far from the far zone of the tape, which is endless. And p will be transfinitely remote from both the far zone and the 0 end. In short you cannot have your cake and eat it. My point is, no transfinitely long tape or tape r/w process proceeding in steps will be actualisable. A tape may loop in a finite span or run finitely in a line but it will not be open ended and transfinite. Or, loop and be transfinite.

    PPPS: Let me symbolise a transfinite loop, the :0* being plugged into the ^0 to loop.

    ^0 . . . EoE . . . H:k+1, k, . . . 2, 1, 0*

  280. You can’t start in the “far zone”. There is no “place” at negative infinity to start. You can inductively move towards infinity, so to speak (an informal way of saying going on endlessly), but you can’t move from infinity back to zero. It makes no sense to speak of “starting in the far zone.” If you want to start someplace a long ways before zero, you still have to start at some finite number.

    So, meaningless question.

  281. KF,

    DS, Let us look at two managers, one who starts at the far zone of the hotel infinity and inspects rooms, one from the near zone. Both set out at one room per second and have to inspect all rooms. Will either ever complete, why? KF

    Aleta is right, there is no such “far zone”. Also, the manager we have been talking about, who is just finishing the tour now, never “set out”.

    If you had simply asked whether a second manager, starting now at the front desk and working backward would ever finish, the answer is no. But again, the situation is not symmetric; one manager began at a specific point, the other one was on a beginningless tour.

    Now, can you answer my simple yes/no question: Are any cells on the infinite Turing Machine Tape infinitely far from the end?

    You can explain all you want, but I’m requesting that you first respond with either “Yes” or “No”.

  282. Aleta & DS,

    Don’t you see that you agree with me by implication?

    Take the model:

    // . . . H:k+1, k, . . . 2, 1, 0.

    At every finitely remote k + 1, k, the H shows that was once the present. But the subscripts allow us to lop off the tail and go:

    // . . . H:k+1, k |

    Obviously this is formally equivalent to (using primes for the new onward k’s):

    // . . . k’+1, k’, . . . 2, H:1, 0 |

    which is just as remote from any infinite past to the left beyond the “break” marks.

    The claim of an infinite past looks to be meaningless, it is reducing to claims of a finite past (of whatever extent) but with an endlessness tacked on that cannot be represented or accepted.

    If there were an infinite causal, temporal past succession of events and entities, the past would have to stretch off leftwards like the negative x axis with an arrow pointing to infinity in the past or off to the left.

    In rejecting that by claiming every past point is finitely remote, that is tantamount to a finite past.

    If every “milestone marker” to the left — pastwards direction — is finitely remote the total increment of necessity will be finite.

    KF

    PS: Same, for a paper tape

  283. KF,

    In rejecting that by claiming every past point is finitely remote, that is tantamount to a finite past.

    If every “milestone marker” to the left — pastwards direction — is finitely remote the total increment of necessity will be finite.

    KF

    PS: Same, for a paper tape

    Ok let’s be very clear. Are you asserting that the infinite Turing Machine tape pictured here has cells which are infinitely far from the end?

    This is a yes/no question.

  284. DS, you tell me what infinity means to you, please. Then, let us see an actual picture or drawing without ellipses or open ended lines or perspectival tapering to a point at the horizon or other vanishing point, of what an actual 0.1 inch pitch 8-wide paper tape would look like. Row 10^150 would be 10^149 inches off or 1.58*10^144 miles from the start with no end of onward rows to come — row k here would be formally equivalently far from the remote zone as row 0. The tape, so far as I can understand, must be endless to be infinite. KF

  285. KF,

    By an infinite Turing Machine tape, I mean one in which the cells are in 1-1 correspondence with the set of natural numbers.

    In the picture I linked to, the leftmost cell is cell 0, the first one to its right is cell 1. Cell n is n steps to the right from cell 0, for any natural n.

    I don’t know what you mean by “remote zone” here, but I will say that if you took two of these tapes and cut off the first 10^150 cells of one of them, they would remain indistinguishable (assuming no symbols had been written in the cells).

    And obviously I can’t draw a picture of the entire tape. Nevertheless, we have been talking about an infinite Hilbert Hotel without any pictures.

    So again, are there any cells in this tape infinitely many steps from cell 0?

    Edit:

    The tape, so far as I can understand, must be endless to be infinite. KF

    Well, in the picture, there is no right-hand end to the tape. Each cell has an adjacent neighbor to its right.

  286. DS:

    Okie, let us compare, showing end to RHS and endlessness to LHS:

    Counting No’s

    N: . . . EoE . . . k+1, k, . . . 2, 1, 0

    Tape Rows:

    T: . . . EoE . . . R_k+1, R_k, . . . R_2, R_1, R_0

    All at 0.1 inch pitch, per standard.

    The relevant part in both cases is. . . . EoE . . .

    Endlessness to LHS means for the tape unending rows at 0.1 in pitch. Set k = 10^150 and it would be 1.58*10^144 miles from the start with no end of onward rows to come, indeed, you could start the count from k (just use the subscripts) and it would make no difference to what is to the L.

    An actual infinite tape would have to be endless to the L, and an actual endless timeline of causal events and entities would be the same, only in time.

    KF

  287. KF,

    I guess you’re talking about an infinite Turing Machine tape with an end on the right but no end on the left? Like in the picture, but with directions reversed? If so, that’s what I have in mind. [Edit: To be clear, my tape consists of just a single row of cells.]

    My question (for the third time) is:

    Ok let’s be very clear. Are you asserting that the infinite Turing Machine tape pictured here has cells which are infinitely far from the end?

    This is a yes/no question.

  288. DS, endlessness is endlessness, it cannot be finite. There is no LH end and the remote zone is infinitely far away, with endless holes 0.1 inch apart all the way. KF

  289. Gotta run now, but is that a “yes”?

  290. DS:

    AmHD:

    in·fi·nite (?n?f?-n?t)
    adj.
    1. Having no boundaries or limits; impossible to measure or calculate. See Synonyms at incalculable.
    2. Immeasurably great or large; boundless: infinite patience; a discovery of infinite importance.
    3. Mathematics
    a. Existing beyond or being greater than any arbitrarily large value.
    b. Unlimited in spatial extent: a line of infinite length.
    c. Of or relating to a set capable of being put into one-to-one correspondence with a proper subset of itself.

    If it is not endless it is not infinite.

    If the far zone is such that every row of holes is finitely many times 0.1 inch away, it is not infinite.

    Begin to see where some of my concerns lie?

    See where oh the set is infinite but every number in it is only finitely large becomes of significant concern?

    See where issues of concept arise?

    Infinite implies boundless, beyond ending, not finite.

    Indeed, meaning 3c above exploits that, as the reason a certain proper subset can be matched 1:1 with the original transfinite set is that both are endless. And with the punched tape illustration, every successive row is a standard 0.1 inches further along. So endless values in succession implies endless distance, giving punch to the meaning.

    If it is not REALLY that, go get your own words, infinite is already occupied.

    KF

    PS: For meaning 3c, try starting the count over from k = 10^150, recognising that this is simply a finite subset capable of being put into 1:1 correspondence with the original set. Think, a pink and a blue tape, only you pull in the blue 10^150 holes (~10^144 miles) and then match it against the pink tape’s 0 end. Endlessness is endlessness, it makes no difference.

    PPS: After you do that, pull in to the 10^300th hole, 10^150 times the first distance . . . use the first pull as a yardstick and do it 10^150 times over. Then put the blue tape in match with the pink — conveniently, holes can be optically lined up. There is still no difference.

    PPPS: Do it over and over again with no end (use an endless loop algorithm), no difference. Endlessness is endlessness.

    P^4S: the observed cosmos is about 5 * 10^23 mi across.

  291. F/N: Collins ED is even better:

    infinite (??nf?n?t)
    adj
    1.
    a. having no limits or boundaries in time, space, extent, or magnitude
    b. (as noun; preceded by the): the infinite.
    2. extremely or immeasurably great or numerous: infinite wealth.
    3. all-embracing, absolute, or total: God’s infinite wisdom.
    4. (Mathematics) maths
    a. having an unlimited number of digits, factors, terms, members, etc: an infinite series.
    b. (of a set) able to be put in a one-to-one correspondence with part of itself
    c. (of an integral) having infinity as one or both limits of integration. Compare finite2

    KF

  292. We know all this. What is your answer to dave’s question?

  293. Aleta, already given and emphatically underscored. If not actually infinitely — endlessly — remote in the far left zone, then finite and not infinite. With the tape going at 0.1 inch per row leftwards. Hence, my conceptual concerns. KF

  294. No, you haven’t “given and emphatically underscored” an answer. You’ve repeated your concerns, but I can’t tell whether your answer is yes or no.

    I’m sorry to have to interrupted Dave’s conversation with you (sorry, Dave), but it is not clear what your answer is because of all the confusion about what you think endless and infinite mean. So Dave’s question is trying to get to some specifics that clarify the concepts:

    Are you asserting that the infinite Turing Machine tape pictured here has cells which are infinitely far from the end?

    Yes or No

  295. Aleta, if infinite does not mean endlessly remote beyond any finite but arbitrarily large value, what does it mean? And, if it does not mean that any actually finite value is not endlessly remote beyond any arbitrarily large finite value — here at 0.1 inch per row, what does it mean? Where, too, if it does not mean that one may repeatedly — an arbitrary number of times, even with an endless loop algorithm — pull in any arbitrarily large but finite range (I picked 10^150 and its square to draw out the point) endlessly but have no effect on the remaining endlessness, what does it mean? KF

  296. Then your answer is No: there are no cells which are infinitely far from the end.

    Is this correct?

  297. Aleta, I have actually gone back to ensure that the opposite is clearly intended. Let me put the just adjusted up again:

    if infinite does not mean endlessly remote beyond any finite but arbitrarily large value, what does it mean? And, if it does not mean that any actually finite value is not endlessly remote beyond any arbitrarily large finite value — here at 0.1 inch per row, what does it mean? Where, too, if it does not mean that one may repeatedly — an arbitrary number of times, even with an endless loop algorithm — pull in any arbitrarily large but finite range (I picked 10^150 and its square to draw out the point) endlessly but have no effect on the remaining endlessness, what does it mean?

    If your conception of the infinite is so radically diverse, why use the same terms? KF

    PS: I add that this includes that if one claims an infinite past of the observed cosmos and its physical predecessors, then one claims a past that is endlessly remote and spanning beyond any arbitrarily large but finite time of the past, e.g. 10^150 s or its square or its square of 10^300 s taken any number of times in succession without scratching the surface of the endlessness, etc.

  298. Must. Not. Ever. Answer. Simple. Question!

  299. So your answer is “Yes”???

    Or is it “No”

    Please, answering the question with a paragraph with lots of rhetorical questions isn’t useful.

    Just answer, with one word, Yes or No.

    Are you asserting that the infinite Turing Machine tape pictured here has cells which are infinitely far from the end?

  300. Aleta (attn HRUN),

    I have been crystal clear and consistent, from laid out sequences of ordinals to dictionaries to concrete examples and on to asking pointed questions in that light.

    If infinite does not mean “endlessly beyond any even arbitrarily large but finite value or things tantamount to that,” then it has been turned into a synonym for finite.

    It comes to a point that if after that degree of emphasis has been put up, inquisitorial yes/no answers are repeatedly demanded, that is a sign that something is very wrong. And not with what I have said.

    Where, it is precisely because the infinite as far as I can reasonably gather means as I have again summarised, that I find something jarring in the claim that per induction the set of naturals has only finite members in it although the set as a whole is transfinite in cardinality.

    Let me add: In terms of the punch tape example, if the endless extension of the tape does NOT have in it rows that are endlessly far from the originating end, something is wrong. So far, that we can take away any arbitrarily large but finite initial range from the tape and it would still be endless beyond.

    In that context I have repeatedly pointed to the importance of the ellipsis of endlessness, and have further noted that the EoE is in the LHS of the assignment of ordinal value: {0, 1, 2 . . . } –> w. (I have even gone so far as to examine the difference between unlimited extension of a chain of inferences per implication from case k to case k +1 in steps from an initial value and spanning the endless. In effect, there seems to be an often unstated but implicit sub axiom of spanning the endless through pointing onwards from a potentially transfinite chain, that is doing a lot of work and carrying a heavy load.)

    KF

  301. KF,

    See where oh the set is infinite but every number in it is only finitely large becomes of significant concern?

    Erm, no, I don’t understand the concern, tbh.

    Indeed, meaning 3c above exploits that, as the reason a certain proper subset can be matched 1:1 with the original transfinite set is that both are endless. And with the punched tape illustration, every successive row is a standard 0.1 inches further along. So endless values in succession implies endless distance, giving punch to the meaning.

    If it is not REALLY that, go get your own words, infinite is already occupied.

    But it is. It’s trivial to see that the cells in the infinite tape can be put into 1-1 correspondence with a proper subset of its cells. We’ve already been over that. Associate each cell to its neighbor on its right (for the tape in the picture).

    I have yet to see a yes-no answer to my question. My best guess is that now you are saying that the thing I’ve been calling an “infinite Turing Machine tape” is not actually infinite??

  302. “Inquisitorial yes/no” questions!?

    OK, here is what I think your answer is.

    There are no cells (numbers) which are infinitely far from the end of the tape (that is infinitely far from zero).

    Infinite means “beyond any even arbitrarily large but finite value”. No matter how long the tape runs (no matter how far we count), we will never be infinitely far from zero. We will always be a finite distance from zero.

    I would agree with both of those propositions.

    Have I stated them in a way that you could agree with?

    If we knew we agreed with each other on this, then maybe it would clear up some confusion.

  303. Aleta,

    I’m sorry to have to interrupted Dave’s conversation with you (sorry, Dave), but it is not clear what your answer is because of all the confusion about what you think endless and infinite mean.

    No problem at all!

  304. KF,

    It comes to a point that if after that degree of emphasis has been put up, inquisitorial yes/no answers are repeatedly demanded, that is a sign that something is very wrong. And not with what I have said.

    When Aleta and I ask for yes/no answers, that’s exactly what we mean. We would like you to literally type “Y-E-S” or “N-O” in your reply, because we have such a hard time figuring out what you are saying.

    At this point, I’m still not clear what your position is. If you would actually answer yes or no, then it would save us a lot of guesswork.

  305. Aleta (& DS), I think the core issue is that something has gone wrong with the meaning assigned to infinite. So, there is a gap of concept at work — which makes simplistic y/n answers meaningless; and, it is one where infinite in certain contexts seems to have been implicitly redefined to mean finite. Endless beyond any finitely, arbitrarily large value has been effectively erased, it seems when it comes to the Naturals. As at now, it seems to me that the way it has been done is based on a use of induction that is open to challenge. As I have I believe reasonably shown from 217 on, no inductive stepwise chain of extension of counting numbers can span the endless. Indeed when it reaches to any finite value k then k + 1 as immediate successor, we may truncate the so far and start afresh and still be at the beginning, indeed putting the onward chain in 1:1 correspondence with the original one. That is what the discussion on pink and blue punch tapes was about. Instead mathematical induction hanging from an initial value and succession logic or steps shows open-ended reliability and points to the potentially infinite but does not actually complete the endless. Claiming or implying ending or spanning or traversing the endless in stepwise succession — even of logical steps of inference — is a fallacy. We have implicitly imposed a sub axiom that pointing onwards suffices. In this case, the succession of counting sets incrementing step by step endlessly, actual endlessness seems to be material. KF

  306. You write,

    [It] points to the potentially infinite but does not actually complete the endless. Claiming or implying ending or spanning or traversing the endless in stepwise succession — even of logical steps of inference — is a fallacy.

    I agree with you.

    a. We can never complete the endless – I agree with this.

    b. Claiming or implying ending or spanning or traversing the endless in stepwise succession — even of logical steps of inference – is a fallacy – I agree with this also.

    We are in agreement.

    I wrote some similar statements:

    c. There are no numbers which are infinitely far from zero.

    d. Infinite means “beyond any even arbitrarily large but finite value”. No matter how far we count, we will never be infinitely far from zero. We will always be a finite distance from zero.

    Do you agree with my statements? They seem to be saying the same things that you are, so do you agree with my statements?

  307. Aleta, please see how I have cleaned up and highlighted. I think there is a significant conceptual issue. I am beginning to think there is an implicit redefining of infinite to mean finite in the context of the naturals. If they are endless, how do we say they are ALL finite apart from taking induction a bridge too far, from open ended reliability to traversing the endless? KF

    PS: Just above c seems jarring and in conflict with d in context.

  308. I agreed with two of your main points in 308.

    I also made two statements which I think are consistent with your position, and asked if you agree with them.

    Do you agree with c. and d. in 308?

    I know there are large issues, but perhaps we could make progress if we figured out what we agree upon.

  309. Aleta:

    c. There are no numbers which are infinitely far from zero.

    d. Infinite means “beyond any even arbitrarily large but finite value”. No matter how far we count, we will never be infinitely far from zero. We will always be a finite distance from zero.

    That our counts or chains of implication taken in steps etc will always be finitely remote from a start point does not in my view imply that the set of counting numbers, as endless, will not go endlessly beyond any arbitrarily large but finite value.

    I am saying we cannot span the set in steps but can point to its endlessness by showing a typical pattern and pointing onward through ellipsis of endlessness.

    I think the two things I just said are consistent with each other, and do not entail that all in-principle counting sets are of finite cardinality. Endlessness of the succession implies or at least strongly suggests to my mind at least that some counting sets in succession will themselves be endless. Those, we cannot reach to by a finite span of steps of extension of the typical counting set that starts from {} –> 0, {0} –> 1, {0,1} –> 2 etc.

    That would seem to be required to get to endlessness [in principle], per the sort of issues in 217 above.

    KF

  310. Can you explain the conflict between c and d. Lets see if we can work together to come up with some language we agree upon.

    Do you agree with c and not d, or vice versa, and could you word either c and/or d in ways that you could agree on?

    Let’s focus on what we do agree on.

  311. KF,

    I am beginning to think there is an implicit redefining of infinite to mean finite in the context of the naturals.

    The set of natural numbers is infinite. It is not finite because it can be put in 1-1 correspondence with a proper subset of itself.

    There are many infinite sets all of whose elements are finite.

    Take the set of all 1/n where n is a positive integer. Every single element in this set is less than or equal to 1 and greater than 0.

    There are finite sets with only infinite elements. {ω, ω + 1, ω + 2}, for example.

    Edit: The cardinality of a set is independent of the magnitudes of its elements.

  312. I was writing 312 when you posted 311, and am interested in narrowing things down to very precise statements, one at a time.

    You write, “That our counts or chains of implication taken in steps etc will always be finitely remote from a start point .”

    So we agree that as we count, we will always be a finite distance from our starting point.

    I agree with that statement.

    Do you agree with that statement?

  313. By definition, one cannot instantiate an infinite set.

    Therefore there cannot be an infinite number of past seconds (or any time interval you choose.)

    Is this really so difficult?

  314. Aleta:

    d. Infinite means “beyond any even arbitrarily large but finite value”. No matter how far we count, we will never be infinitely far from zero. We will always be a finite distance from zero. –> AGREED, IN CONTEXT OF COUNTING AS A STEPWISE PROCEDURE OR THE CHAIN OF SUCCESSIVE IMPLICATIONS IN ORDINARY MATHEMATICAL INDUCTION, ETC.

    c. There are no numbers which are infinitely far from zero. –> NO, THIS DOES NOT FOLLOW. NOT IN A WORLD OF ELLIPSES OF ENDLESSNESS WITHIN SETS THAT CONTAIN THE COUNTING SET SUCCESSIONS.

    KF

  315. DS, Yes, the set 1/n is transfinite, it is a successive chain of rationals in the interval [0,1] i.e, an approach to the infinitesimals next to 0. The steps of extension are not finite all the way — another use for infinitesimals. This is the same issue Zeno faced, and obviously this set is of same cardinality as n; and [0,1] is itself transfinite of order beyond aleph null. But with the counting sets the succession to endlessness is automatically enfolded in successive members and the increments are finite and divergent not convergent, there is no delta neighbourhood of a limit where beyond some point for any delta range all onward elements will be within delta of the limit. Where, we cannot actually span the endlessnes but we point to it. KF

  316. M62, unfortunately, this is obviously not so simple. And I am acutely aware I am swimming upstream of conventional wisdom, due to concerns that point to at minimum paradoxes. This stuff is also freighted with pretty heavy potential worldview implications tied to the infinite past some claim, infinitesimals and calculus, the nature of numbers and more. KF

  317. KF,

    The steps of extension are not finite all the way.

    Well, the steps are actually positive all the way.

    1/n – 1/(n + 1) = 1/(n*(n + 1)) is never 0 nor infinitesimal.

    In any case, I don’t think there is any issue of redefining infinite to mean finite in the context of the natural numbers.

  318. Good – this is progress. Summary: we agree about the following in respect to counting in step-wise fashion:

    a. We can never complete the endless.

    b. Claiming or implying ending or spanning or traversing the endless in stepwise succession is a fallacy.

    c. No matter how far we count, we will never be infinitely far from zero. We will always be a finite distance from zero.

    Good.

    d. The next step is to consider the statement whether there are numbers infinitely far from zero:

    I, Aleta, say “There are no numbers which are infinitely far from zero.

    Kf says “NO, THIS DOES NOT FOLLOW. NOT IN A WORLD OF ELLIPSES OF ENDLESSNESS WITHIN SETS THAT CONTAIN THE COUNTING SET SUCCESSIONS.”

    So, two new questions:

    Q1: Do you agree that we have reached some clarity about what we agree on, and that we have an issue, d above, to address?

    Q2: Also, are you willing to continue to try to take things one or two statements at a time, continuing to search for agreement when we can?

  319. DS, what is the limit of the sequence {1/n} as n increases without limit, i.e. endlessly . . . and notice this routine use of endlessness in Math. I suggest to you that for any arbitrarily small delta neighbourhood of 0, the error [1/n – 0] = epsilon will be below delta for all SEQUENCE terms beyond some k, the particular k being dependent on how small delta is. Effectively k ~ 1/delta, actually the next whole number beyond it. That is the limit is zero and the size of 1/n diminishes without limit other than zero as n becomes endlessly large. KF

  320. Aleta, I believe Q1 is yes, Q2 is it depends as there may be concept gaps, I won’t go all the way to paradigm shift but that is related. KF

  321. KF,

    The limit is zero, but all pairs of consecutive numbers in the sequence are a positive distance from each other and from zero.

    That’s not a controversial statement at all.

  322. DS, did the force of the epsilon delta relationship in the neighbourhood of 0 as n increases endlessly make my point clear? The gap between 1/n and 1/(n + 1) goes to infinitesimal levels as n increases without limit. Note, that use of endlessness. 1/10 vs 1/11 is about 10% different 1/1,000 to 1/1,001 is 1/10 of 1% different already, and it goes on from there. think about the difference between 1(10^300) and 1/(1 + 10^300), then go onward without limit. KF

  323. Ok, next:

    320 c, which we agreed on, says “No matter how far we count, we will never be infinitely far from zero. We will always be a finite distance from zero.”

    Q1 for this post: Could we rewrite that in conditional form: if a number can be reached by step-wise counting, it is finite”?

    Q2: Is the converse true: if a number is finite, it can be reached by step-wise counting?

    So, is Q1 an acceptable rewrite of a point we have agreed on, and do you think the converse in Q2 is true, or not?

  324. KF,

    No, it doesn’t go to “infinitesimal levels”, ever.

    1/n – 1/(n + 1) is always a positive real number.

    I assume we both know what a convergent sequence is, so maybe we should leave this issue for now and concentrate on N.

  325. DS, your remark is tantamount to you stay in a finite band of n. That is exactly what the limit is not. KF

  326. Aleta, a number REACHED by stepwise counting is finite and if finite it can be reached by a long enough count chain. The issue is not the finite neighbourhood of 0 out to some arbitrarily large k, k + 1 succession. No, it is that having hit k, k + 1 etc, we can throw everything away so far and put the onward count in 1:1 correspondence with the original count precisely because the onward count is endless. KF

  327. KF,

    DS, your remark is tantamount to you stay in a finite band of n. That is exactly what the limit is not. KF

    I defined the sequence as a_n = 1/n for each positive integer. Each positive integer is finite.

    The sequence does not contain any terms 1/n where n is non-finite. Yet it converges to 0.

    All this is completely standard.

  328. Good, kf, you combined the two statements into one, which I think could be stated as

    “A number can be reached by counting if and only if it is finite”

    That is,

    all numbers reachable by counting are finite, and

    any finite number is reachable by counting.

    I’m pretty sure that is all in agreement with what you wrote: “Aleta, a number REACHED by stepwise counting is finite and if finite it can be reached by a long enough count chain. ”

    So this is progress. Now I’m going to think about your next sentence in 328, about what the issue is.

    And I really appreciate your willingness to take this step-by-step.

  329. You write, “[the issue] is that having hit k, k + 1 etc, we can throw everything away so far and put the onward count in 1:1 correspondence with the original count precisely because the onward count is endless.”

    I’m not sure I understand this, so let me offer an example and you can tell me whether I am correct or not.

    Consider the set S1 = {1, 2, 3, 4, 5, …}

    Now we “throw away everything so far” and consider the “onward count” S2 = {6, 7, 8, 9, …}

    S1 and S2 can be put in a 1:1 correspondence because both sets are infinite, since both counts are endless.

    Is this an example of what you mean?

    If not could you give an example?

  330. Aleta, In the blue and pink tape examples, I set k = 10^150, and noted that a k, k+1, . . . onward sequence would match 1:1 with the original from 0, 1, 2 . . . (never mind discarding 10^144 miles of blue tape) due to endlessness. So, if k = 8 much the same would follow. KF

    PS: A number reached or reachable by step by step finite stage counting processes will be finite, but an endless set of numbers cannot be exhausted by such counting processes. The counting process will only be potentially infinite. The endlessness will exceed count processes and linked chaining processes such as inference case k => case k+1. At some stage, the sub axiom of pointing to and filling out the ellipsis of endlessness in one step of generalisation will be called in. In the case of N, it is with a divergent sequence that is endless. Where as the paper tape illustration shows, endlessness has significance. If the tape is endless, how can its far zone only be finitely far away from the near end in 0.1 inch steps? (And, does not the claim, all counting numbers are finite but the set of such as a whole is endless not then pose at minimum a paradox?)

  331. DS, the set of positive integers is by definition endless. Endlessness counts, as I have pointed out. KF

  332. KF,

    DS, the set of positive integers is by definition endless. Endlessness counts, as I have pointed out. KF

    I’m not saying that the set of positive integers has an “end”, i.e., a last or greatest element.

    I am saying that in the sequence I defined, the differences between consecutive terms is always positive and never “infinitesimal”. Likewise, 1/n is positive and not “infinitesimal” for all positive integers n.

    Edit: Wait, I just read #332. Are you saying that the set of positive integers, being “endless”, therefore has infinitely large members??

  333. Thanks kf:

    1. I seem to understood correctly concerning the issue of “throwing everything away so far.

    2. I also assume that we agree that both S1 and S2 in my example are infinite sets, because S2, a proper subset of S1, can be put in 1:1 correspondence.

    3. You write in 332, and I agree. We can consider this a settled point.

    A number reached or reachable by step by step finite stage counting processes will be finite, but an endless set of numbers cannot be exhausted by such counting processes. The counting process will only be potentially infinite.

    4. You then write, and this is where I’d like to start today,

    At some stage, the sub axiom of pointing to and filling out the ellipsis of endlessness in one step of generalisation will be called in. In the case of N, it is with a divergent sequence that is endless. … If the tape is endless, how can its far zone only be finitely far away from the near end in 0.1 inch steps?

    You’ve mention the sub axiom of the ellipsis of endlessness (EoE) several times, and also that of a far zone: it is these ideas that I don’t understand.

    It seems to me that you are saying that there are numbers in the set that are infinitely far away from the starting point as well as the numbers that are finitely far away.

    Question: Is this accurate? Is this what you are saying?

  334. DS,

    Let us take the finite series terms:

    Term n = SUM (1^i) from i = 1 to n,

    Then, let n increase without limit, colloquially, increase endlessly, go to infinity.

    What happens to term n?

    What does this tell us about the span of the naturals?

    KF

  335. KF,

    The sequence of partial sums for the original sequence a_n = 1/n diverges, of course.

    Edit: Have to go for now, but I’m not sure what your sum actually is in #336. I thought you were talking about summing 1/1 + 1/2 + 1/3 …, but it’s not clear now.

    I don’t know what you mean by the “span of the natural numbers”, but of course the set of positive integers contains no largest member.

    To reply to your post #333 again,

    DS, the set of positive integers is by definition endless. Endlessness counts, as I have pointed out. KF

    If a necessary condition for a set to be “endless” is that it contains infinite members, then no, the set of positive integers is not “endless” in this sense.

  336. DS, pls cf 336. KF

  337. See my edit to #337. What’s the sum again?

  338. 336

  339. Well, that’s not terribly helpful. Here’s how I interpret #336:

    Term n = SUM (1^i) from i = 1 to n,

    1^i = 1 for all integers 1, so we have:

    SUM(1) from i = 1 to n

    which is simply n of course.

    So Term n = n for all positive integers, and this sequence diverges.

    Something tells me you don’t mean to add up a bunch of expressions of the form 1^i, however; why put a useless exponent on the 1?

    Would you please tell me what the sum is, exactly?

  340. DS, I exactly meant each term n to add up to n exploiting 1^i = 1. Now, what of the term n when n goes up without limit, what used to appear as the infinity symbol sitting on top of the sigma? KF

  341. Bumping 335:

    From 335:

    You’ve mention the sub axiom of the ellipsis of endlessness (EoE) several times, and also that of a far zone: it is these ideas that I don’t understand.

    It seems to me that you are saying that there are numbers in the set that are infinitely far away from the starting point as well as the numbers that are finitely far away.

    Question1 : Is this accurate? Is this what you are saying?

    Also, I’ve been thinking about what “far zone” might mean.

    Question 2: Is the far zone a region that contains numbers?

  342. KF,

    DS, I exactly meant each term n to add up to n exploiting 1^i = 1. Now, what of the term n when n goes up without limit, what used to appear as the infinity symbol sitting on top of the sigma? KF

    Well, as I stated above, the sequence a_n = n diverges.

    I don’t know what this is supposed to tell me, however.

  343. Aleta,

    Observe how a step by step chain of counting or implication at any finite stage k –> k+1 can readily be started afresh, and will be just as far from the exhaustion of the ellipsis of endlessness as though we were starting from 0; indeed the 1:1 correspondence technique applies to the fresh labels, just go k –>0, k+1 –> 1 etc. But, routinely, such are extended through the ellipsis as though that were a single step that sweeps the board. Such seems to be typically a sub axiom, often implicit. As what bridges the ellipsis, it is doing the heavy lifting and is a focus for concern. Again, the endlessness is pivotally important. KF

    PS: Go to the tapes with 0.1 inch pitch for concreteness. If there is endlessly remote tape there will be rows of appropriate rank, and that raises the point they should be beyond any finite scale away. As, finite is tantamount to not endlessly remote. Thus my concerns.

  344. Hi kf. First, I agree with you when you write, “the endlessness is pivotally important.” The nature, meaning, and consequences of endlessness is what I’d like to discuss.

    You write,

    If there is endlessly remote tape there will be rows of appropriate rank, and that raises the point they should be beyond any finite scale away

    Question: Does the phrase “they should be beyond any finite scale away” answer my question in the affirmative: yes, there are numbers that are infinitely far away?

    That is, you answer Yes to the question “are there numbers in the set that are infinitely far away from the starting point?” Am I correct that you answer yes to this question?

  345. The funniest part of this thread is taking place over on TSZ where keiths said there could be a finite set that keeps growing and growing forever that wouldn’t be infinite.

    Seriously, he said that.

  346. Aleta, if the rows are all finitely distant, it would appear that they cannot at the same time be endlessly — infinitely or better, transfinitely — distant. Thus, the concern. And again that one step through the ellipsis of endlessness seems to be carrying a lot of weight. KF

  347. VC, KS may be expressing phenomena tied to Hilbert’s Grand Hotel Infinity. Endlessness cannot be traversed in finite steps, and at any particular point the stepwise process will have only attained a finite extent, ever pointing onwards to the potentially infinite — as opposed to the actually completed infinite. I am suggesting that if that is his meaning it would be much as the case in 217 above. KF

    PS: I have seen someone point out that the eschatological kingdom would be of that character, potentially infinite in the mathematical sense but enduring ever after unto ages of ages without end. Indeed, here is Dan 2: “44 And in the days of those kings the God of heaven will set up a kingdom that shall never be destroyed, nor shall the kingdom be left to another people.” Note, endlessness. And of course all of this started with claims about an actually infinite spatio-temporal past for the physical cosmos we inhabit and its antecedents. I again suggest if the endless cannot be traversed in steps one way, it cannot the other way either. So the most plausible view is a finite past to our world. And yes that points to ultimate beginning and to cause.

  348. re: 348. I am trying to understand the concern. However, it is confusing to me when you make replies that don’t clarify the questions I’ve asked.

    Obviously, all the rows that can be reached by steps are finite – we’ve agreed to that.

    What I am asking is this: are there actual numbers that are “endlessly distant”?

    In 345, you said, “If there is endlessly remote tape there will be rows of appropriate rank, and that raises the point they should be beyond any finite scale away.”

    Assuming the pronoun “they” refers to “rows”, this seems to say that some rows will be “beyond any finite scale away”, which I assume means they would be an infinite scale away.

    So I am not sure whether you are saying that there are, or that there are not, numbers an infinite (endless) distance from the starting point.

    Therefore, the question:

    Are there numbers an infinite distance from the starting point

    I understand you have a concern. Answering this question will help clarify the concern.

  349. Aleta,

    My thought is, on endlessness there SHOULD be transfinitely remote rows in an endless tape, or equivalently that the endless succession of incrementing counting sets should at the remote zone attain to the transfinite. I use zone to emphasise endlessness and nonspecificity. The ellipsis of endlessness is again pivotal.

    And if all such rows or successive counting sets in sequence are in fact finite, then it would seem that the extent must be finite.

    I have a suggestion (now with a spot of adjustment for further specificity):

    unlimited cumulative stepwise processes based on finite stages point to the endless and manifest the potentially infinite. This includes ordinary mathematical induction as well as counting set chains and processes that are tied to such. On conceptually traversing the ellipsis of endlessness (often implicitly), we contemplate the ideal set of counting numbers as a whole we cannot actually exhaust; we assign this whole the order type w and the cardinality of first degree endlessness, aleph null. Where, the endlessness is in the LHS:

    {} –>0, {0} –>1, {0,1} –> 2, . . .

    {0,1,2 . . . k, k+1, . . . EoE . . .} –> w

    Could this be a way forward?

    It seems to capture my thought and points to my concern.

    Notice, this implies a distinction between open-ended chaining of induction and actually exhausting the ellipsis of endlessness.

    KF

  350. re 351: interesting and useful comments.

    You write,

    My thought is, on endlessness there SHOULD be transfinitely remote rows in an endless tape, or equivalently that the endless succession of incrementing counting sets should at the remote zone attain to the transfinite

    Although “remote zone” is left undefined, it seems that this formulation says that the natural numbers, as defined by N = {0, 1, 2, 3, …} contains both finite numbers, which can be reached by steps, and, by virtue of the ellipsis, also transfinite numbers which cannot be reached by steps, but nevertheless exist out of reach.

    Your argument for this is in the next paragraph,

    And if all such rows or successive counting sets in sequence are in fact finite, then it would seem that the extent must be finite.

    That is, there can’t be an infinite set of numbers all of which are finite.

    I don’t believe this formulation is mathematically viable in that it could not be rigorously defined. I believe this formulation is trying to grasp a real and important notion, endlessness, but that concepts such as “beyond the ellipsis” and “remote zone” are more metaphors for the result of endlessness than they are mathematical ideas that could ever be properly formulated.

    The part that you quote after “I have a suggestion” may help me explain further, and may lead to some common understanding.

    You write,

    stepwise processes point to the endless and manifest the potentially infinite. This includes ordinary mathematical induction as well as counting set chains and processes that are tied to such. On conceptually traversing the ellipsis of endlessness (often implicitly), we contemplate the ideal set of counting numbers as a whole we cannot actually exhaust; we assign this whole the order type w and the cardinality of first degree endlessness, aleph null.

    I think the part I bolded is a key. To help explain, I want to distinguish between the process of endlessness and the result.

    In respect to the natural numbers, the ellipsis refers to the unending nature of the process: we can always take a next step. This is concrete and can be formulated mathematically with precision.

    However, when we try to conceptually grasp the result of that unending process, we go beyond what we can mathematically formulate within the framework of the natural numbers themselves: when we, to use your nice phrase, “contemplate the ideal set of counting numbers as a whole we cannot actually exhaust,” we are stepping “up a level”, so to speak, in our conceptualizing.

    Even though mathematicians and others have been contemplating the unending infinite for a long time, Cantor formalized it in the way you state: “we assign this whole the order type w and the cardinality of first degree endlessness, aleph null.” That is, we take the concept of the result of unendingness and give it a name: as I described in a post on the history of math, we create a new kind of number, starting with aleph null, and from there build a whole new branch of mathematics concerning transfinite numbers.

    So here is one way to think about resolving the issues you have brought up: a way that doesn’t involve “traversing the ellipsis” nor any “remote zone”.

    We can look at either endlessness as process or we can look at the result of endlessness as the concept of an infinite set.

    Endlessness as a process is what the ellipsis inside the natural numbers means. When we write {1, 2, 3, …}, the ellipsis refers to the process by which this set can be built endlessly. The ellipsis does not, however, refer to the result of the process.

    The result of the process, the “ideal set of counting numbers as a whole we cannot actually exhaust” is described with a number (not an ellipsis) that is not in the natural numbers. Aleph null, stands above, so to speak, the natural numbers, and formalizes our concept of an ideal set – one which is infinite. It is aleph null which contains the resolution of your concern by encapsulating the infinite nature of the result of the unending process expressed by the ellipsis.

    This distinction helps me, at least, understand how to resolve the sense that you describe that there has to be something else “out there”, remotely and transfinitely beyond the finite naturals that can be reached in steps. The “remotely out thereness” of the set of natural numbers – there infinite nature, is dealt with by the mathematics of transfinite numbers, but it doesn’t include anything inside the set of natural numbers. All there is in the set of natural numbers are finite numbers, and an unending process. The result of that process – an infinite set – is captured by the transfinite number aleph null.

    Probably this explanation will not allay your concerns. However, I am convinced that the intuitions you have about what is “beyond the ellipsis” could never be mathematically formulated. And to summarize, I think you are trying to squeeze concepts into the natural numbers that don’t need to be there, and are adequately resolved by the created of the transfinite numbers to symbolize and work with the “ideal set of counting numbers as a whole.”

  351. Hi kf. I’ve had some further thoughts that clarify some things for me. The central idea is that the ellipsis itself is an ambiguous symbol that can be interpreted two ways, and that the tension caused by this ambiguity is possibly the source of some of your cognitive dissonance and sense of paradox.

    Building on the distinction I made in 352 between process and product, I see now that possibly the ellipsis itself can be interpreted to mean either process or product.

    (In fact, Wikepedia says, “The use of ellipses in mathematical proofs is often discouraged because of the potential for ambiguity. For this reason, and because the ellipsis supports no systematic rules for symbolic calculation, in recent years some authors have recommended avoiding its use in mathematics altogether.”)

    So here is my idea:

    I can think of two ways to interpret the ellipsis, and this corresponds to the distinction I made in 352.

    You can think of the ellipsis as standing for the rule that creates the natural numbers: given any k, there is a k + 1. This is what I mean in saying that the ellipsis stands for the process. It means “keep on going, following the pattern.”

    However, and occurred to me after writing 352, instead one could think of the ellipsis as standing for all the remaining members of the set. That is, it could represent the entire set of numbers not hitherto enumerated. With this interpretation, the ellipsis would stand for the entire infinite result of the process and not the process itself.

    It may be that this ambiguity is part of what is confusing. The ellipsis as process only produces finite numbers, and at any moment only a finite number of them. The ellipsis as product interpretation includes the entire infinite set.

    So like the Gestalt faces/vase I referenced in a earlier post, flipping back and forth between the two ways to interprete the ellipsis might be the source of your cognitive dissonance. With one view, you only see a finite number of elements, and with another view you see an infinite number. However, because the process/product distinction isn’t clear, your attempt to see both meanings at once, with an emphasis on the product interpretation, leads to your sense that there must be something more, something infinite, in addition to the finite number produced by the step-wise process.

  352. Aleta

    There is some closeness here.

    In 217 I laid out an endless loop algorithm that shows the stepwise process of advance which cannot exceed the potentially infinite. Not least, that at any k, k+1 pair, the whole can be started again as though it were 0. Just promote the subscripts.

    Loop after loop as an unending unlimited process short of pulling the plug.

    This immediately implies that ordinary mathematical induction is unable to transit to the actually exhausted transfinite.

    It can show that we have a potential infinite and that to any actual value we can attain stepwise or write down

    [which is tantamount to the same; as, the place value number writing system

    . . . p*b^k+1 + q*b^k . . . *b^1 +s*b^0 + t*b^-1 + u*b^-2 + . . .

    (b, the base in use — often 10 but sometimes 2, 12, 20 or 16 or 8 or 60 etc )

    . . . is a finite in fact potentially infinite power series, note its own ellipses of endlessness]

    it will hold by chaining: that Claim-X hanging on Claim-X, case 0 or 1 by successive implication also holds. However, that is in itself a limitation of the potential infinite.

    The ellipsis carries a lot of weight, and there is the implicit resort to a conceptual step across the transfinite span of the ellipsis, what I have called a sub axiom that is carrying a lot of weight.

    Contemplating the ideal, completed set that we cannot actually exhaust by stepwise process, we then assign it an order type w and a cardinality of endlessness in the first degree, aleph null.

    Through the ordinals, we revert to the same over and over again, and when we reach an uncountably large . . . not even countable in principle . . . one w1, we assign this the cardinality aleph 1, generally found to be the power set of aleph null.

    Then we continue again.

    The result of all this is there is a serious problem in how we tend to think about the naturals.

    The process of counting and pointing to the potential endlessness does not exhaust the endless, it only points to it.

    Then, when we go to ordinary mathematical induction, we go: here is claim C-0 or C-1, and here is a proof that for any pair k, k+1 C-k => C-k+1. Then we play the magic step of the ellipsis of endlessness and voila we say this pervades the endless. Process is imagined complete and the whole is said to have been swept in just one further conclusive step.

    This relies on what I have called an implicit sub-axiom.

    That is, the spanning of the ellipsis of endlessness by pointing.

    This carries a lot of weight and is ambiguous between unending process and endlessness. I can see the reason for the deprecation, but I wonder how ever so many things in mathematics, science and engineering can get on without the ellipsis in maybe a tamed form.

    What this leads me to hold as a plausibly reasonable view for the moment . . . yes, I am emphasising provisionality . . . is that:

    in looking at the chain of counting sets as laid out in a von Neumann type construction, and in applying ordinary mathematical induction to this, we have only really addressed what an endless loop algorithm can do. That is, we have an in principle endless procession of cumulative finite stage steps. This is utterly distinct from claiming to have exhausted, or actually completed or ended the endless. Instead, we must recognise the ambiguity, and we must reckon that anything reachable by a stepwise finite stage process will by definition be finite and bounded by a successor. However such a loop process cannot exhaust endlessness. Thus, we are forced to make a conceptual leap — by way of implicit sub axiom — to an ideal view of a completion we cannot attain stepwise, and we then proceed beyond the ellipsis of endlessness. In this case, we look at how, on conceptually traversing the ellipsis of endlessness (often implicitly), we contemplate the ideal set of counting numbers as a whole that we cannot actually exhaust; we assign this whole the order type w and the cardinality of first degree endlessness, aleph null. Where, the endlessness is in the LHS:

    {} –>0, {0} –>1, {0,1} –> 2, . . .

    {0,1,2 . . . k, k+1, . . . EoE . . .} –> w

    What this says to me, is that we are dealing with a gap that is unbridgeable by ordinary stepwise processes.

    This means to me that we have no right to say that all potential counting sets in succession of actual endlessness are of finite scale, but we can freely say that all counting sets actually reachable by processes based on the stepwise approach will be finite. Finite, because we can always show them bound by going on one more step in some fashion.

    Once we move to the inherently abstract contemplation of the potential endlessness, we then can assign the whole ideal process of attaining the end of the rainbow an order type omega and a cardinality, the first degree of endlessness, aleph null.

    It seems that we are finite and bounded in ways we cannot even readily imagine. But at the same time, we can contemplate the ideal world, here we can dare say, the form of the endless and transfinite.

    The ghost of Plato is laughing.

    Coming back, we see that there is always a problem when one claims or implies traversing the endless in a stepwise, inherently finite process.

    Which, to my view, would include claims about an actually endless and now completed to present, causal chain

    . . . EoE . . . Ck+1 –> Ck –> . . . –> Cn, now.

    Ending the endless through a stepwise, inherently finite process of successive, cumulative chaining is a fallacy.

    As an endless cycling of the wheels on a vehicle gripping a road can only ever attain a finite distance — as, one can always drive on a bit further — so also, stepwise, finite stage cumulative processes cannot span the transfinite.

    Which can be shown by cycle counting an endless loop algorithm. Cycle k always leads to cycle k+1 and we can promote the ellipsis and in effect start over again, no closer to exhausting endlessness than we were before.

    There is a verse in the grand old revival hymn, Amazing Grace:

    When we’ve been there ten thousand years…
    bright shining as the sun.
    We’ve no less days to sing God’s praise…
    than when we’ve first begun.

    Strikes me, there’s some’at in that.

    KF

  353. PS: This then opens up the issue of the closed interval [0,1], the infinitesimals next door to 0, and the catapult hyperbolic function 1/x that pushes such infinitesimals out to the transfinite zone, starting with the Hyper Reals and Hyper Integers. The issue then begs for an answer, how are such related to the naturals and reals, ordinals and transfinites as we have explored. Above, I have played with some speculative models of exploration and they suggest that if we define mild infinitesimals m, we can project to the transfinite zone and create a continuum there between w +k and w+ (k+1) etc, where the ordinals by direct analogy to the real number line and the naturals, will be like milestones. Hard infinitesimals h, we can then project onwards out to the zone of the continuum cardinality, whether that be aleph 1 or some successor or whatever. But this is little more than playing with ideas, it is part and parcel of stirring the pot, an explicit theme of this thread of discussion. If this is feasible and coherent on serious analysis, I cannot say, but I can say that the playing around in a sandbox for a few moments is interesting — as it was interesting for me as a child with a real sandbox. And it suggests the further point that we need to bring together the various models of the transfinite zone and clarify their relationships.

  354. F/N: I only very rarely look at TSZ, but just did so on a Feb 1 KS post that of course sneers at the benighted fundies that cannot get their heads around how a set with all members finite and separated by finite stages can be transfinite or endless. Evidently, there is a failure to see that finitude inherently speaks to boundedness and the infinite to boundlessness or endlessness. I suggest to KS, there is an issue of the ellipsis of endlessness and there is a difference between a process that being stepwise and cumulative on finite stages can only ever be bounded — For k has k +1 to follow — and the endlessness it points to. That is there is a difference between the potentially and actually infinite and even an endless loop that takes finite steps cannot span the ellipsis of endlessness. We are forced to take a sub axiom of completing the span by pointing, and address the emergent phenomenon of endlessness in its own terms. That is w succeeds the ellipsis, not any particular finite member of the set that collects successive counting sets aka the naturals. It is a limit ordinal, as Wolfram reminds and as was mentioned above. KF

    PS: SalC, pardon my taking up your argument as it aptly illustrates my point about the sub axiom being implicitly used:

    Let n = 1, then n is finite since there exist a number which is greater than n, namely n+1.

    By induction we show all natural numbers are finite since there exists for every n, a natural number greater than n, namely n + 1.

    For there to be a natural number n+1 for every n in the naturals implies the set of naturals is infinite.

    This first chains, not recognising that chaining in steps in an unlimited loop will only progress in steps. So, at any pair n, n+1 we reset by promoting the subscripts and it is as though we have only just begun. We are chasing the end of the rainbow which is always unreachably far ahead of us.

    Ordinary mathematical induction is inherently dependent on finite step, stepwise cumulative chaining and does not in itself bridge the endless, as say we see in 217 above.

    So, what is done is to implicitly bring in the ellipsis of endlessness and span it in one step.

    The Reddit discussion has the same pattern, jumping from n to n+1 then spanning endlessness in one onward leap.

    Yes, the set that collects successive, cumulative counting sets is inherently endless. That is the point.

    And inferring from the inherent finitude of cumulative steps that the endlessness as a whole only contains the finite as members becomes paradoxical. The sub axiom of the long jump across the endless is carrying the weight of the conclusion.

    And when one premise carries all the weight, that is a point that should give us pause. Circularity beckons. So does the problem of ending the endless in one swooping conceptual leap.

    Hence, my concern.

    PPS: I see the point that there is a reasonable bridge between information and entropy is still a problem. (Cf April 10 2015 thread here.) I simply note that again, and observe there is a paradigm shift. Please see Harry S Robertson on Statistical Thermophysics. I point to the discussion in Thaxton et al, TMLO chs 7 – 9 and to the observation that there is a difference between clumping from scattered state and functional organisation on configuration to specific plan. One may speak of work to clump and onward work to configure, and to relevant entropy reductions, which as this is a state function can be done in principle all at once. The point remains since 1984 that it is maximally implausible for such ordering and organising to functionally specific pattern to occur by blind watchmaker forces. And, as a personal note on tone, I have no need to prove to you re my qualifications, which you know to be legitimately acquired.

  355. I have to go to work today, (I’m mostly self-employed), but I’ll say this:

    You write,

    in looking at the chain of counting sets as laid out in a von Neumann type construction, and in applying ordinary mathematical induction to this, we have only really addressed what an endless loop algorithm can do. That is, we have an in principle endless procession of cumulative finite stage steps. This is utterly distinct from claiming to have exhausted, or actually completed or ended the endless. Instead, we must recognise the ambiguity, and we must reckon that anything reachable by a stepwise finite stage process will by definition be finite and bounded by a successor. However such a loop process cannot exhaust endlessness. Thus, we are forced to make a conceptual leap — by way of implicit sub axiom — to an ideal view of a completion we cannot attain stepwise, and we then proceed beyond the ellipsis of endlessness.

    More or less agree, if you take “beyond the ellipsis of endlessness” to be a metaphor for contemplating the set as a whole in the manner of Cantor. However, if “beyond the ellipsis of endlessness” means that within the natural numbers there is some “far zone” that is beyond the reach of the finite numbers created by k > k + 1, then no.

    Unless such “far zone” can be defined with some mathematical precision, and I don’t believe it can, I think your feelings about it are really about the mystery of trying to intuit an infinitely large set. And since we really can’t intuit the infinite, the thing to do is fall back on what mathematics tells us, as per Cantor and transfinite mathematics.

  356. kairosfocus:

    VC, KS may be expressing phenomena tied to Hilbert’s Grand Hotel Infinity. Endlessness cannot be traversed in finite steps, and at any particular point the stepwise process will have only attained a finite extent, ever pointing onwards to the potentially infinite — as opposed to the actually completed infinite.

    There isn’t any such thing as “the actually completed infinite”. Infinity is a journey, a never-ending journey.

  357. KF,

    Just out of curiousity, have you read the wikipedia page on the Peano Axioms carefully? If you haven’t, I think it would answer quite a few of your questions, for example why an infinite Turing Machine tape need not have any cells infinitely many steps from the single end. (Not that this is inconceivable, but it’s not necessary and is certainly not how people typically think of these things).

    Here’s part of the section on the Axiom of Induction, which reads:

    To do this however requires an additional axiom, which is sometimes called the axiom of induction. This axiom provides a method for reasoning about the set of all natural numbers.

    If K is a set such that:

    * 0 is in K, and

    * for every natural number n, if n is in K, then S(n) is in K,
    then K contains every natural number.

    The simple proof of the finitude of all natural numbers that Tao wrote has K = the set of all finite natural numbers, and because K satisfies both starred hypotheses, K is all of N. Hence every natural number is finite.

    You’ve expressed doubts about whether mathematical induction can accomplish such a thing, but the fact that it can is actually an axiom. And if you’re going to discuss N, then that axiom is in the “rule book”, so to speak.

  358. For clarity, here’s a better-formatted version of the axiom of induction:

    If K is a set such that:

    * 0 is in K

    and

    * for every natural number n, if n is in K, then S(n) is in K

    then K contains every natural number.

  359. DS:

    Do you see the sub-axiom at work?

    If K is a set such that:

    * 0 is in K, and

    * for every natural number n, if n is in K, then S(n) [–> the successor to n] is in K,
    then K contains every natural number.

    The point here is that what we can properly do is case 0 or case 1 depending, and the chaining claim c-k => c-k+1, which then chains in an endless loop. That is unlimited but it cannot exhaust endlessness and looks at taking the leap to the end of the rainbow.

    The sub-axiom is carrying the heavy weight, again.

    And for every n begs the question, what is n and how do we get there.

    KF

  360. Aleta, Beyond means taking the set as a whole in one step and in context of the ordinal sequence putting up w as successor. KF

    PS: When I look at the vases faces gestalt image I can see a superposition of the two interpretations, especially when colours other than B/W are used, and I can switch mode to rapid alternation of the two views too. Long time ago now, a pastor used to talk of how when John used an ambiguous word strategically he often means both senses as a subtle wordplay of superposition and mutual mirroring to send a richer yet sense — double meanings on steroids I suppose. Q-bits anyone?

  361. VC, that is a heart of the issue, there is no stepwise, finite stages, cumulative attainment of an actually completed infinite. We point to the potential and sweep across an ellipsis of endlessness. KF

  362. KF,

    DS:

    Do you see the sub-axiom at work?

    ***

    The point here is that what we can properly do is case 0 or case 1 depending, and the chaining claim c-k => c-k+1, which then chains in an endless loop. That is unlimited but it cannot exhaust endlessness and looks at taking the leap to the end of the rainbow.

    No, there is no endless loop in the proof of “if n is finite, then n + 1 is finite”.

    Take an arbitrary natural number n > 0. Then n = {k ∈ N | k < n}, which is a finite set, so no “EoE” here. BTW, I’m using the von Neumann definition of ordinals here, as you have throughout this discussion; the finite ordinals coincide with N.

    But then n + 1 = n ∪ {n}, another finite set, so n + 1 is finite.

    And for every n begs the question, what is n and how do we get there.

    n is a natural number. It is either 0 or the successor of a natural number. The axioms describe exactly what they are and how we “got there”.

  363. DS, n to n +1, repeat. KF

  364. KF,

    No, the relevant part of my proof above consists of 3 sentences, with no “looping” construction or repetition.

  365. DS, you don’t see the recursiveness involved, aka endless looping? KF

    PS: Wiki on Peano: “Axioms 1, 6, 7 and 8 imply that the set of natural numbers contains the distinct elements 0, S(0) [successor to 0], S(S(0)), and furthermore that {0, S(0), S(S(0)), …} [subset or equal to] N”

  366. KF,

    No, there is no recursion or endless looping in the proof of the statement “for all natural numbers n, if n is finite, then n + 1 is finite”.

    Here’s a simple example that may be less confusing: * (for every n) if n is an even integer, then n + 1 is an odd integer.

    To prove this, let n be an even integer, so n = 2k for some integer k. Then n + 1 = 2k + 1, which is odd.

    This proves statement *, with no recursiveness or endless looping.

    I agree with the wikipedia quote, by the way.

  367. DS, if we start from 0 is in N then say for each n in N S(n) its successor is in N, that is recursive. And the successive process will lead to a finite set of bounded numbers — which by being exceeded in turn will be finite [and notice how the looping emerges as the successor has its own etc] — but cannot exhaust the full set which is endless. It points to it as a potential but not completed infinity of process. Cf the von Neumann type establishment from {} –>0, {0} –> 1, {0,1} –> 2 etc. This shows a potential infinity and as per usual, goes to the fulness of the set with that ellipsis. KF

  368. KF,

    Of course, the set of natural numbers is defined inductively. That’s why I wonder how you make sense of the entire “completed” set N in the first place.

    However, the proofs of “for all natural numbers n, if n is finite, then n + 1 is finite”, and “for all integers n, if n is even, then n + 1 is odd” are not recursive or inductive.

    Edit: Your post above illustrates what I said above. I don’t think you can speak of a “completed” set N.

  369. F/N: Decided to glance back at TSZ, just to see. I was not disappointed:

    1: Note to KS:

    The context is there (let me add by way of emphasis: regrettably), on much backstory by you and ilk. With Dawkins’ atrocious sneering remark about ignorant, stupid, insane or wicked ever lurking.

    I also add: there is need to focus the issue. And the infinite turns out to be far more subtle of an issue than appears.

    2: Note to Dazz:

    keiths: The natural numbers truly can be exhausted by a stepwise process. It just takes an infinite number of steps.

    [Dazz] That’s what I’ve always understood too.

    An infinite or endless process is of course exactly what cannot be completed in finite stage cumulative steps. Fallacy of ending the endless.

    After every k, k+1 steps we can throw away what has gone before promote the subscripts to full counts, and act as though this is the beginning, and be no closer to ending the endless, and that can be done in an endless loop to the same effect. Above I set k = 10^150 then its square 10^300. Makes no difference.

    What we can do is do something enough to show a potential infinity then point onward with an ellipsis and act on that ideal, that vision, that mental abstraction, that proposition.

    And maybe that is the problem, do you accept the difference between the contemplative mind and the rock that can have no dreams?

    KF

  370. DS, I see we can define numbers in a way like von Neumann, and from the succession so established, point to potential infinity and accept the set as a whole as endless. That then leads to cardinality of first degree endlessness and order type w. From N, Q, from these R, thence C thence ijk vectors and structures etc. KF

  371. KF,

    Well, in any case, are we agreed that the above proof of the statement “for all integers n, if n is even, then n + 1 is odd” contains no infinite loops or repetition?

  372. DS, it seems to me the endless loop is right there in “for ALL . . . ” esp. if you see how the set is built up from {} by succession. As in how do you get TO evens plural much less an endless span of them. KF

  373. VC, I should footnote on your two counters, one counting s the other every 2nd s. These both go to potential infinity, but indeed one is slower. Think of the two punch tapes, pink and blue. Now add a third in that ugly beige we all remember from IBM punch cards. This one, however has an oddity, rows are 0.2 inches apart. If all three are endless, does this have as many holes as the other two? The answer being, the numbers are endless. And if the blue is pulled in 10^150 holes and the ~10^144 miles of tape is cut off, all three remain endless. Aleph null stands in for first degree endlessness that can be matched 1:1 endlessly (note, process not completion) with the natural numbers. But in any finite equal length beige will have about half the number of rows as pink and blue. The issue is endlessness. KF

  374. F/N:

    Over at TSZ, KS tries a clincher:

    If you accept that the naturals are constructed upward from 0 (or 1), like this…

    0 is a natural number.

    If n is a natural number, then n+1 is a natural number.

    …and if you accept the following premise…

    If n is finite, then n+1 is finite.

    …then it follows that every natural number is finite.

    It’s that simple.

    If you disagree, then where does the argument go wrong?

    The proof of course is familiar, and my concern with it is the same. Ending the endless, and perhaps even making the transfinite mean the finite.

    The general process is:

    {} –> 0
    {0} –> 1
    {0,1} –> 2

    . . .

    {0,1,2 . . . (k-1)} –> k
    {0,1,2 . . . k} –> k +1

    . . .

    The sub axiom of the ellipsis of endlessness appears in its usual ghostly three dot garb, and we move from a step by step loop that inherently cannot actually complete the endless to inferring to the completed endless.

    Cannot actually complete?

    Yes, at a given n = k, then k+1, we are in effect able to cut off what has gone before and start over 0, 1, 2 . . . so that at no finite k are we ever close to completing the endless.

    Which means, no finite number is endless, and no span of finite numbers will be endless.

    Go back to pink and blue punch tapes, starting here and going on endlessly. Pull in the blue k steps, say 10^150, i.e. ~ 10^144 miles. There are still endlessly many rows ahead.

    For any finite k, the span of tape is 0.1 inch x k, an inherently finite value. So if all the values of n are finite, then the tape is finitely long at any k. For ALL values of k. Which is problematic if one is claiming actual endlessness.

    This is the sort of context in which I am concerned with claims of an infinite, actually endless succession of counting numbers, n.

    What I suggest is, we set up endless loop processes that point to the potentially transfinite, though in themselves they are finite. Always.

    The finite is not the transfinite.

    Endlessness is pivotal and the ellipsis of endlessness is key.

    This also extends to ordinary mathematical induction. For it endlessly chains from a claim, case 0 or 1. C-0 or C-1 and C-k => C-k+1, then go cases 2,3 . . . and there we see the ellipsis of endlessness that cannot be spanned in steps.

    In these cases what we are doing is we take the implicit step of achieving an endless loop based potential infinite then infer completion of the endless as an ideal concept.

    So, we go:

    {0,1,2 . . . EoE . . . } –> w

    and so forth.

    And those are where my concerns lie on claiming that

    all n in the set just shown are finite,

    as opposed to,

    all we can count to or actually represent (say as in place value notation decimal or binary form, which are disguised power series of form: . . . p*b^k+1 + q*b^k . . . *b^1 +s*b^0 + t*b^-1 + u*b^-2 + . . . –> note the chained, successive subscripts in the whole number and fractional parts) will be finite.

    KF

  375. KF,

    DS, it seems to me the endless loop is right there in “for ALL . . . ” esp. if you see how the set is built up from {} by succession. As in how do you get TO evens plural much less an endless span of them. KF

    If there were any kind of loop in the proof, I would have to start with some particular even integer n, show that n + 1 is odd, then move to another even integer and repeat. None of that appears in the proof. I dealt with all even integers simultaneously.

  376. KF,

    Regarding your post #376:

    all we can count to or actually represent … will be finite.

    Let K be the set of all natural numbers we can count to. Clearly 0 is an element of K, and if n is in K, so is n + 1.

    Therefore by the Axiom of Induction, K = N. That is, every natural number is a number we can count to, so every natural number is finite.

  377. DS, the loop is there. And the pointing onward across the ellipsis of endlessness is also there. I add, including in the axiom of induction. Wolfram: ” If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.” KF

  378. KF,

    Here’s the proof:

    Let n be an even integer, so n = 2k for some integer k. Then n + 1 = 2k + 1, which is odd.

    Can you name the first five even integers I’m looping over?

  379. DS, you are using a process which rests on a looping process reflected in simply the n itself. I am not saying that this skips over integers, but that it is inherently based on endless looping with what that entails for stepwise processes addressing the endless. KF

  380. KF,

    So does the following summarize your position on this question?

    My proof shows that for any finite subset S of Z, if n ∈ S is even, then n + 1 is odd.

    It does not show that for all n ∈ Z, if n is even, then n + 1 is odd.

  381. DS, I am pointing out that the inference to the scale of Z is based on stepwise, endless loop inference which is then extended through an endless ellipsis. And that the successor of an even will be an odd is very different from the import of in an endless set of successive counting sets, all the sets will be finite. KF

  382. KF,

    Well, the correct analog to “the successor of an even will be an odd” is “if n is finite, then n + 1 is finite”. Both of which I accept of course. Both seem equally trivial to me.

    I’m interested in what we can prove. You have stated that we cannot prove that all natural numbers are finite (assuming the Peano axiom system, say). Correct?

    On the other hand, you say we can prove that if S is a finite subset of N, all the elements of S are finite. Again, correct?

  383. DS, that in a stepwise succession pattern n being finite entails n +1 finite is not the same as addressing endlessness; a stepwise iterative, cumulative, finite stage process will not span the endless. And it is the transition to endlessness which is where my concerns lie as has been stressed over and over. Ending the endless, explicitly or implicitly, is an error. KF

  384. KF,

    It’s possible that we’re just not interested in the same things then. I’m focusing on what can be proved in a particular axiom system, as well as the issue of “beginninglessness”, as WLC describes it, illustrated by the reverse HH tour and the eternal ticking clock.

    I haven’t said much about “ending the endless”, and I don’t think that occurs in mathematical induction. At least so far as I can understand without a rigorous definition.

  385. DS,

    What has been of secondary concern to me is the logic of the system of numbers, how it is operationally defined by step by step succession, and the issue of claiming a finite stepwise process based on stages that are finite, creates a system in which all members are finite but the whole is infinite. That is at minimum paradoxical.

    The primary context has been the claim being made or implied, of an endless spatio-temporal causal succession coming down to now, which interfaces with the mathematics.

    However, as mathematics is in effect the logical study of structure and quantity, logical considerations clearly obtain. One of these is that to claim to have traversed the infinite by cumulative, finite stage steps or to imply such, is to assert a contradiction.

    The endless is not ended by such a stepwise proces.

    Such processes based on unlimited looping of steps are indeed capable of pointing to endlessness, but are themselves finite, only potentially infinite.

    Hence the use of ellipses of endlessness.

    But it is easy to persuade ourselves that the sub axiom of pointing across the ellipsis of endlessness and drawing a conclusion on the whole has settled a matter. The first concern is, doing so unexamined. The second, is that an implicit step (as it so often is) should be viewed with caution when it is carrying a LOT of weight. Third, in the case of inferring that all counting sets in endless succession are finite but the whole is infinite, there is at minimum a paradox.

    In the case of ordinary induction, the problem is as outlined in 217 above and as was repeatedly pointed out. When one sets out a claim say C-j then hangs it on case 0 or 1, then adds that C-k => c_k+1, one is going in an open-ended loop, applying an endless loop algorithm to create a potential infinite then pointing across an ellipsis of endlessness. What is actually established is that we have a claim that will succeed as far as we can reach or enumerate specifically, but we cannot ever actually end the endless. This is where the problem comes out when this procedure is used to infer that say all counting sets built up in succession and continued endlessly are nonetheless finite. In fact the implication of endless building up of successive sets — notice the recursion and do forever loops — is that there is a zone where the sets have in them an ellipsis of endlessness. That is the succession process points to endlessness within individual sets even as the overall collection is extended without limit across its own ellipsis of endlessness.

    Follow the logic to see how that happens.

    Endlessness appears in the successive members as entangles with endlessness of the whole based on the potential infinity of the successor process based on do forever.

    We cannot actually complete, but we look beyond to the idealised:

    {0,1,2 . . .} –> w

    The ellipsis of endlessness carries a LOT of weight.

    We also come full circle to that evolutionary materialist cosmologies face the unhappy choice between pulling being out of a non-existent hat, non-being; or else, ending the endless.

    That is on top of self referential, self falsifying incoherence as regards the capability of mind, which undermines the whole scheme of thought.

    KF

  386. VC 358, yes, the transfinite as we study it is based on do forever loops that we cannot actually complete but point beyond. KF

  387. DS, do you recognise that algorithms or implied ones based on do forever loops cannot actually be extended forever, and so we are forever pointing across ellipses of endlessness to draw idealised conclusions? KF

  388. PS: Note . . .

    http://whatis.techtarget.com/d.....dless-loop

    >>An infinite loop (sometimes called an endless loop ) is a piece of coding that lacks a functional exit so that it repeats indefinitely. In computer programming, a loop is a sequence of instruction s that is continually repeated until a certain condition is reached. Typically, a certain process is done, such as getting an item of data and changing it, and then some condition is checked, such as whether a counter has reached a prescribed number. If the presence of the specified condition cannot be ascertained, the next instruction in the sequence tells the program to return to the first instruction and repeat the sequence, which typically goes on until the program terminates automatically after a certain duration of time, or the operating system terminates the program with an error.

    Usually, an infinite loop results from a programming error – for example, where the conditions for exit are incorrectly written. Intentional uses for infinite loops include programs that are supposed to run continuously, such as product demo s or in programming for embedded system s.

    A pseudo-infinite loop is one that looks as if it will be infinite, but that will actually stop at some point.>>

    Wiki:

    >>Looping is repeating a set of instructions until a specific condition is met. An infinite loop occurs when the condition will never be met, due to some inherent characteristic of the loop.
    Intentional looping

    There are a few situations when this is desired behavior. For example, the games on cartridge-based game consoles typically have no exit condition in their main loop, as there is no operating system for the program to exit to; the loop runs until the console is powered off.

    Antique punch card-reading unit record equipment would literally halt once a card processing task was completed, since there was no need for the hardware to continue operating, until a new stack of program cards were loaded.

    By contrast, modern interactive computers require that the computer constantly be monitoring for user input or device activity, so at some fundamental level there is an infinite processing idle loop that must continue until the device is turned off or reset. In the Apollo Guidance Computer, for example, this outer loop was contained in the Exec program, and if the computer had absolutely no other work to do it would loop run a dummy job that would simply turn off the “computer activity” indicator light.

    Modern computers also typically do not halt the processor or motherboard circuit-driving clocks when they crash. Instead they fall back to an error condition displaying messages to the operator, and enter an infinite loop waiting for the user to either respond to a prompt to continue, or to reset the device>>

    In many systems such loops are used in the main subroutine that then branches on inspected conditions or interrupts. Power down is an interrupt of highest priority and should trigger safe shutdown routines.

    Actually completing an endless process is a fallacy.

    KF

  389. KF,

    I get the “infinite loop” aspect of inductive proofs, but I think you’re not taking into account the fact that we can apply the Axiom of Induction.

    To illustrate, let P be the predicate “is finite”, defined on N.

    Then P(0) = True of course.

    But P(0) = True implies that P(1) = True, so P(1) = True.

    P(1) = True implies P(2) = True, so P(2) = True.

    and so forth. If I proceed in that manner, I will be executing an infinite loop, and at every step, I will have shown only that a finite number of natural numbers are finite. Since a proof is supposed to consist of finitely many steps, this won’t work to prove all natural numbers are finite.

    That’s not how the real proof goes, however. It has these parts:

    1) Show that P(0) = True.

    2) Show that for all natural numbers n, if P(n) = True, then P(n + 1) = True.

    3) Conclude that for all n in N, P(n) is True, by the Axiom of Induction.

    #1 and #2 can be proved in a finite number of steps, as I showed above. Incidentally, in this case #2 is proved without induction, but there would be nothing wrong with having to do a second round of induction at this stage.

  390. KF,

    DS, do you recognise that algorithms or implied ones based on do forever loops cannot actually be extended forever, and so we are forever pointing across ellipses of endlessness to draw idealised conclusions? KF

    Well, I’m not sure. Why couldn’t a “do forever” loop be extended forever? It wouldn’t be taking place in space and time, so I don’t know what the barrier would be.

    On the other hand, that’s emphatically not what is happening in the proof that all natural numbers are finite. The Axiom of Induction takes care of that.

  391. DS,

    please read 217 above to see how the endless loops you keep putting up critically depend on pointing across an ellipsis of endlessness to move to the idealised concept of a completed infinite; always involving endless iterative loops starting with constructing the set of counting sets itself.

    Notice, a telling little phrase in 391: “and so forth” . . . i.e. a signature of a do forever loop.

    To suggest an actual end of the endless via finite stage steps, is a fallacy. We can get to a potential infinite but we cannot actually complete it.

    In the case of cumulative creation of counting sets, at each stage we extend one level in succession. It is reasonable to infer from this that if the process is endlessly repeated the accumulation inside the sets will also become that. Endless, a synonym for transfinite.

    The answer to which is, we never actually complete such a process, indeed to end the endless would be a contradiction, so we project an ideal. But also that means that the proof by ordinary induction has a gap between what is actually shown and the conclusion that goes beyond that.

    Yes, if C-0 holds and C-k => C-k+1 then we may do forever, but that cannot actually be completed, only envisioned. At any k, k+1 pair we may proceed to put in parallel with the original and start the count over again [exploiting that property of the transfinite which is of defining character], never getting closer to ending the endless.

    What we show is the cumulative process will always be finite and bounded as the next step is a bounding cap-off of where we actually reached so far. That is an opposite character to endlessness. Going further, we show that as far as we can reach or represent with place value numbers [which involve power series also] the general claim particularised as so for case 0 will obtain.

    We then apply a sub axiom of pointing across the ellipsis of endlessness and often conclude to for ALL cases. Often that works as the pattern does not rely on the cumulative effect of endlessness.

    But in the case of the counting numbers themselves, every step extends so if they are extended endlessly, perforce the counting sets at some zone should be just that, endless.

    We may workaround by imposing a general successor to the endless set and stipulate that w is the first transfinite, but the simple accumulation of successive counting sets points to if extended endlessly, individual members should tend to become endless, in effect copies of the set as a whole. If on the other hand the members are all determined to be finite, the set as a whole, which reflects the progress of its members, should remain finite also.

    Do forever cannot actually be completed though it may be continued unendingly. Note the difference and the stress on continuation, pointing to a gap between the potential and the actual infinite.

    Pointing across the ellipsis of endlessness is doing the heavy lifting, and is in this case pointing to a paradox.

    Hence, my concern.

    I think we have gone a bridge too far.

    KF

  392. KF,

    Notice, a telling little phrase in 391: “and so forth” . . . i.e. a signature of a do forever loop.

    Yes, but as I stated, that’s not the real proof of the finitude of all natural numbers.

    If you’re still thinking that the proof consists of:

    P(0) = True
    If P(0) = True then P(1) = True, therefore P(1) = True
    If P(1) = True then P(2) = True, therefore P(2) = True

    and continuing on in an infinite loop, that’s incorrect.

    See the three steps in the second blockquote for the steps in the real proof.

  393. DS:

    Again, notice the implied succession:

    1) Show that P(0) = True.

    2) Show that for all natural numbers n, if P(n) = True, then P(n + 1) = True. [–> here comes the do forever represented by an endless loop]

    3) Conclude that for all n in N, P(n) is True [–> as in point across the ellipsis of endlessness], by the Axiom of Induction.

    Where, on Axiom of induction, per Wolfram, we see the same pattern of reasoning:

    Induction Axiom

    The fifth of Peano’s axioms, which states: If a set S of numbers contains zero and also the successor of every number in S [–> do forever, represented by ellipsis of endlessness], then every number is in S.

    I have no problem with defining a set by endless succession, save that we need to recognise what that is doing and that it may have subtleties when we move to the far zone implicit in going across an endless span.

    As pointed out earlier, the stepwise succession of counting sets if we go do forever to endlessness will imply that successive elements in the far zone move to being embedded copies of the endless set. Something we can see by setting up a 1:1 match.

    We may impose some sort of point across the ellipsis, close the double bracket and say order type is w, transfinite, of cardinality first degree endlessness i.e. aleph null, but the heavy lifting is in the ellipsis on the LHS:

    {} –> 0

    {0} –> 1

    {0,1} –> 2

    . . .

    { 1, 2 . . . } –> ?

    . . .

    or, as 217 above illustrated a week ago:

    {0, 1, 2 . . . { 1, 2 . . . } . . . ) –> w

    (BTW, in turn we see subnesting of the same push to endlessness in members of the far zone.)

    And, S has in it every number in an endless context of do forever is suggestive . . .

    Again, paradoxes and concerns.

    KF

  394. KF,

    2) Show that for all natural numbers n, if P(n) = True, then P(n + 1) = True. [–> here comes the do forever represented by an endless loop]

    No, no looping involved:

    Let n be an arbitrary natural number.

    P(n) = True implies the ordinal n is a finite set.

    But then n + 1 = n ∪ {n}, also a finite set.

    Therefore P(n + 1) = True.

    This simultaneously proves P(n) = True implies P(n + 1) = True, for all natural numbers n.

    If you still maintain there is a do forever loop in that proof, kindly write out the loop explicitly.

    Induction Axiom

    The fifth of Peano’s axioms, which states: If a set S of numbers contains zero and also the successor of every number in S [–> do forever, represented by ellipsis of endlessness], then every number is in S.

    Let’s be clear: Are you saying that proving a nonempty set of natural numbers is closed under the successor operation always requires a do forever loop?

  395. KF,

    One more question about these claims of do forever loops. It concerns a matter different from finitude, but I think it will be clarifying.

    Let P be the predicate defined on the set of all integers by:

    P(n) = True iff n^3 + 3n^2 + 2n is divisible by 6.

    Do you believe that every proof of the fact that P(n) = True for all integers n must contain a do forever loop?

  396. DS, the do forever looping is there, right in the definitions, constructions of the counting numbers and axioms. Leaving such implicit by rephrasing or redirecting attention does not change that. The issue is endlessness and what it entails, which leads to my concerns. Onwards, it is quite clear no endlessly stepwise finite stage process can actually traverse the endless. Getting to the potentially infinite and pointing across the ellipsis of endlessness needs to reckon with whether that has a relevant and perhaps unexpected impact — esp. if we have a divergent sequence. KF

  397. Hi Dave. I’ve been thinking about this, and here is a fairly formal proof I came up with.

    To prove: P(n) = n^3 + 3n^2 + 2n is divisible by 6.

    Lemma: 3(n + 1)(n + 2) is divisible by 6

    Proof of Lemma: either (n + 1) or (n + 2) must be even, because every other natural number is even. Assume n + 1 is even, so that n + 1 = 2m, for some m. (If n + 2 is even, just use it instead of n + 1)

    Then 3(n + 1)(n + 2) = 3•2m•(n + 2) = 6m(n + 2), which is obviously divisible by 6.

    Proof of main proposition:

    First note that P(n) = n(n + 1)(n + 2), by factoring.

    P(0) = 0, which is divisible by 6.

    Now, assume P(k) for some k is divisible by 6, and then consider P(k + 1):

    P(k + 1) = (k + 1)(k + 1 + 1)(k + 1 + 2) = (k + 1)(k + 2)(k + 3).

    Now consider the difference D between P(k + 1) and P(k):

    D = (k + 1)(k + 2)(k + 3) – k(k + 1)(k + 2) = (k + 1)(k + 2)(k + 3 – k) = 3(k + 1)(k + 2), which is the expression in the lemma, and is divisible by 6.

    Therefore, for all k, if P(k) is divisible by 6, P(k + 1) is greater then P(k) by some number divisible by 6, and therefore P(k + 1) is also divisible by 6. (This argument could easily be written out more formally, but I won’t bother.)

    Therefore, since P(0) is divisible by 6, and for every k, if P(k) is divisible by 6 so is P(k + 1), by induction P(n) is divisible by 6 for all n.

    Q.E.D.

    Any other straightforward proof?

  398. KF,

    DS, the do forever looping is there, right in the definitions, constructions of the counting numbers and axioms.

    Have you read up on the Axiom of Infinity?

    From the wikipedia page:

    This construction forms the natural numbers. However, the other axioms are insufficient to prove the existence of the set of all natural numbers. Therefore its existence is taken as an axiom—the axiom of infinity. This axiom asserts that there is a set I that contains 0 and is closed under the operation of taking the successor; that is, for each element of I, the successor of that element is also in I.

    That takes care of the existence of the set N in one step, no infinite loops involved.

    Did you find any do forever loops in my proof above or that of the divisibility-by-6 statement?

    If not, I don’t think we actually have any examples of such things on the table.

    As I mentioned above, mathematical proofs are typically presented in the context of a first-order theory. Proofs in these formal systems consist of a finite number of steps. From the wikipedia page on first-order logic, with some bolding added:

    A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

    If these do forever loops that you refer to actually existed, it would mean that the so-called “proofs” in which they were embedded were not really proofs at all, which would be quite problematic. I don’t think it’s plausible this monumental oversight could persist for on the order of a century until finally being revealed on an ID blog.

  399. Aleta,

    Very nice proof. That’s essentially what I had in mind, although I think you can carry it out without using induction, as long as you accept some basic number theory facts. The basic idea is that, as you pointed out, in the factorization n(n + 1)(n + 2), at least one of the factors is even and exactly one is divisible by 3, so this is definitely a multiple of 6.

  400. KF,

    Further to my post #400, see this pdf which describes how the Axiom of Infinity can be used to prove the existence of the set N, with no do forever loops, of course.

  401. Oh duh – what you explained is so much simpler. However, when I tested some numbers, like P(3), P(4), and P(5) I noticed that each result grew by a number that was a multiple of 6, and I had proof by induction on my mind, so I went the way I did.

  402. Aleta,

    And I was highly motivated to avoid induction, for obvious reasons!

  403. Yes, I see. You’ll notice I’ve dropped out of the conversation – I reached some interesting understandings for myself, but gave up on kf.

    On the other hand, I was thinking that seeing some actual mathematics in this thread might be nice. One of our points is that kf doesn’t formulate any actual math to support his intuitions and concerns, nor does he respond to any proofs.

    In fact, I thought this was a great paragraph from you:

    If these do forever loops that you refer to actually existed, it would mean that the so-called “proofs” in which they were embedded were not really proofs at all, which would be quite problematic. I don’t think it’s plausible this monumental oversight could persist for on the order of a century until finally being revealed on an ID blog.

  404. Aleta,

    Thanks for the compliment. Your posts have improved my understanding of the issues a great deal.

    One thing that has become more clear to me is that the usual domino analogy for induction can be quite misleading.

    It seems to suggest (to me, anyway) that proofs by induction really amount to applying modus ponens over and over again, ad infinitum, when that’s absolutely not the case. And I think it’s a root cause of the difficulty we’re having here.

  405. DS, briefly the ax of inf is tied to the von Neumann construction, whereby the successor of x is defined as x union {x}. If x is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set, thence the frequently shown train of counting sets:

    there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

    Succession follows and looping. As is embedded in several elements of the set-up.

    KF

  406. KF,

    Succession follows and looping. As is embedded in several elements of the set-up.

    This is false.

    The axiom states that there exists a set I such that ∅ ∈ I and for all x ∈ I, x ∪ {x} ∈ I. End of story, with no looping.

    Do you recognize that you’re claiming to have discovered a disastrous flaw in elementary set theory that has gone undetected for 100+ years?

  407. The definition essentially creates the whole infinite set all at one time: it doesn’t have to be created one element at a time through an infinite number of steps. I think that perhaps is the key confusion.

  408. Re #408:

    Do you recognize that you’re claiming to have discovered a disastrous flaw in elementary set theory that has gone undetected for 100+ years?

    daveS, I am pretty certain that this argument carries zero weight here at UD. 🙂

  409. That’s right, Aleta.

    I will qualify my #408 by saying again that there are a few mathematicians who don’t accept the Axiom of Infinity, and for them, this discussion is moot. I don’t think KF claims to reject the AoI though.

  410. hrun0815,

    daveS, I am pretty certain that this argument carries zero weight here at UD. 🙂

    Heh. Yeah, I guess somebody had to discover this problem.

  411. KF,

    Anyway, after this detour into the Axioms of Induction and Infinity, do you still stay that the statement “All natural numbers are finite” is not provable in ZFC? (And by “all natural numbers”, I mean every single one, not just all natural numbers that have been generated so far, or something along those lines).

    More concretely, do you still say that the infinite, one-ended Turing Machine tape must have cells infinitely many steps from cell 0?

  412. Aleta & DS (attn HRUN), please look again. Several axioms do embed an implied iterative endless loop that creates successors, then a point past the ellipsis of endlessness to bridge to the whole. And it is not any novelty to point to the difference between potential and actual infinities. The very way the set of whole counting numbers is listed and how it is often explained often shows this:

    {0,1,2, . . . }

    {} –> 0
    {0} –> 1
    {0,1} –> 2
    . . .

    KF

    PS: Put up the two tapes, pink and blue, both being endless from a row 0. Endless implies that there are rows beyond any arbitrarily high specific value we can count to or write down. Where to write down in place value notation, we are resorting to a disguised power series with the same underlying issue. Pull blue in 10^150 rows, 10^144 miles at 1/10 inch per row. It is still endless beyond. Do, again and again, same result. Same match 1:1 to the pink again and again,just promote the k, k+1

  413. F/N: Collins Eng Dict

    >>infinite (??nf?n?t)
    adj
    1.
    a. having no limits or boundaries in time, space, extent, or magnitude
    b. (as noun; preceded by the): the infinite.
    2. extremely or immeasurably great or numerous: infinite wealth.
    3. all-embracing, absolute, or total: God’s infinite wisdom.
    4. (Mathematics) maths
    a. having an unlimited number of digits, factors, terms, members, etc: an infinite series.
    b. (of a set) able to be put in a one-to-one correspondence with part of itself
    c. (of an integral) having infinity as one or both limits of integration. Compare finite2

    ?infinitely adv
    ?infiniteness n>>

    KF

  414. CED:

    >>finite (?fa?na?t)
    adj
    1. (Mathematics) bounded in magnitude or spatial or temporal extent: a finite difference.
    2. (Mathematics) maths logic having a number of elements that is a natural number; able to be counted using the natural numbers less than some natural number. Compare denumerable, infinite4

    3.
    a. limited or restricted in nature: human existence is finite.
    b. (as noun): the finite.
    4. (Grammar) denoting any form or occurrence of a verb inflected for grammatical features such as person, number, and tense
    [C15: from Latin f?n?tus limited, from f?n?re to limit, end]
    ?finitely adv
    ?finiteness n>>

    I take sense 2 to speak of specific number, and that the definitions are tantamount to saying that the set of naturals is a restriction to the finite, but the succession is unlimited.

    Once we have an unlimited, endless succession, the issue is there.

    KF

  415. PS: Notice, I am speaking to a framework and its context, not oh every conceivable point is this and if it is not my concern falls to the ground. That scattershot rhetorical strategy is irrelevant. It is clear there is a step of completing a potential infinite by taking endless loop steps and pointing onward, and it comes up over and over again. In that context, I long since put up the two punch tape reels that go on endlessly in a one-sided infinity, and I again point to the implications of endlessness of such a tape. The issue seems, is endlessness really endless or not? If not, it is finite.

  416. KF,

    Aleta & DS (attn HRUN), please look again. Several axioms do embed an implied iterative endless loop that creates successors, then a point past the ellipsis of endlessness to bridge to the whole.

    No looping is implied. As Aleta stated, the Axiom of Infinity essentially creates the set N all at once. The set is abstract. Why do we need to resort to looping when the set can be created in one fell swoop?

    PS: Put up the two tapes, pink and blue, both being endless from a row 0. Endless implies that there are rows beyond any arbitrarily high specific value we can count to or write down.

    You might be using your own peculiar definition of “endless” here, but I merely stated that the tape was infinite. It is true that you can repeatedly shift the blue tape 10^150 cells and realign it with the pink tape, but you haven’t demonstrated the existence of any cells infinitely far from cell 0 in either tape.

    All you’ve shown is that the natural numbers can be put into one-to-one correspondence with the natural numbers greater than or equal to any multiple of 10^150. That’s consistent with every cell in either tape being a finite distance from cell 0.

    Regarding the dictionary definitions, they are all consistent with every natural number being finite and all the cells on the tape having finite distance from cell 0.

    In that context, I long since put up the two punch tape reels that go on endlessly in a one-sided infinity, and I again point to the implications of endlessness of such a tape. The issue seems, is endlessness really endless or not? If not, it is finite.

    It has infinitely many cells. Here’s the illustration I have been using.

    We can number the cells using von Neumann ordinals.

    The leftmost cell is cell ∅

    The next one, its “successor”, if you will, is cell {∅}

    After that, {∅, {∅}}.

    If I point to any cell in the tape, and tell you its number/ordinal, then you can use a purely mechanical process to find the number of its successor.

    There is no rightmost cell in the tape, because it’s infinite.

    However, each cell is labeled with a finite ordinal.

    There is no bound to the distances between cells and cell ∅. Given any arbitrarily large finite ordinal n, there is a cell n in the tape.

    The set of cells can be put into 1-1 correspondence with proper subsets of itself, which is what your pink/blue tape illustration shows.

    That summarizes my understanding of the infinite Turing Machine tape.

  417. KF,

    Reading this more carefully:

    PS: Put up the two tapes, pink and blue, both being endless from a row 0. Endless implies that there are rows beyond any arbitrarily high specific value we can count to or write down.

    I think this can be interpreted in two very different ways:

    1) For every natural number n, there exists a cell C beyond cell n. That is correct.

    2) There exists a cell C such that for every natural number n, cell C is beyond cell n. That is incorrect.

    Are we agreed on that?

  418. Dave writes,

    I think this can be interpreted in two very different ways:

    1) For every natural number n, there exists a cell C beyond cell n. That is correct.

    2) There exists a cell C such that for every natural number n, cell C is beyond cell n. That is incorrect.

    Very good, Dave. I think kf is asserting that 2 is true. I would like to see him clearly affirm or deny statement 2.

  419. One more thought: kf has consistently used vague and undefined phrases such as “going past the ellipsis” into the “far zone.” In doing so, he seems to imply statement 2. It would help add some mathematical specificity and clarity to his position if he would affirm or deny that statement 2 is, or is a part of, what he means by “going past the ellipsis” into the “far zone.”

  420. Aleta & DS (attn HRUN):

    First, per the cited, infinite means endlessly beyond arbitrarily large but bounded thus finite quantities, values or numbers. Thus the significance of the ellipsis of endlessness.

    As I have explicitly cited a second time, yesterday.

    So, DS, kindly stop ascribing idiosyncrasies to me in the face of understandings sufficiently commonplace to have made it into standard dictionaries.

    Please, Aleta, look again at endlessness.

    That which is represented by the ellipsis of endlessness. Then compare the implications of the way in which the set of counting functions is established and how ordinary mathematical induction is set up by chaining.

    Induction is the key example. Case C-0 or sometimes C-1, and C-k => Ck+1, thence for all cases that we can chain to or denumerate using say decimal place value numbers (which imply power series representations). A potentially infinite is set up, and an ellipsis of endlessness is posed.

    In the case of the natural, counting sets assigned to the chain of usual symbols for numbers:

    {0,1,2, . . . }

    {} –> 0
    {0} –> 1
    {0,1} –> 2
    . . .

    All of this has been pointed out already.

    Consistently, there is what I have called a sub axiom, pointing across an ellipsis of continuation to an endless zone of continuation beyond any given arbitrarily large stated counting number. Or better, at any large k we may succeed by k+1 and then set up a substitution as though we were starting over at 0, 1, 2 . . . i.e. we have the ability to exploit endlessness by setting the proper subset fromk on in 1:1 correspondence with the original set and it will match endlessly. Which is the operational meaning of the infinite.

    This is the context of my having spoken above of the pink and blue punched paper tapes with blue pulled in 10^150 rows and again setting it in matched order with the pink.

    Endlessness is pivotal and the ellipsis is carrying the heavy load.

    If there are no rows in the ordered succession that are not endlessly remote, beyond any arbitrarily large but bounded specific value we care to name, k, then the pull in 10^144 miles and restart the count which will match 1:1 still, would fail.

    So, does endlessness mean endlessness, or does it in reality mean finite but very large? If the latter, then there is a problem of implying ending the endless.

    I do not for the moment care as to how definitions of sets are set up, the issue is the succession to endlessness of rows which are tantamount to counting, ordinal numbers.

    If there are endlessly many rows at 0.1 inch pitch, there are endlessly many rows. And endlessly many numbers in succession beyond our ability to count or represent without resort to the ellipsis of endlessness. Where, such endlessness means there is a far zone that is endlessly far — infinitely far — away. And by that force, there will be ordinal values that are endlessly remote, often denoted by me as w, w+1 and so forth; w stands in for omega. But that does not introduce something that was not there on the LHS:

    {0,1,2, . . . k, k+1, k+2 . . . EoE . . . } –> w

    So also:

    0,1,2, . . . k, k+1, k+2 . . . EoE . . . w, w+1, . . . EoE . . .

    Now, we may denote that w is the first transfinite ordinal, and then make an assignment that the representable ordinals in succession to the EoE beyond any arbitrarily large but specific k etc will be the natural numbers. Then we may make an argument that per ordinary mathematical induction any such number we may specifically represent by k is finite as bound by k+1 etc. However, that still has not eliminated the ellipsis of endlessness and the implied do forever loop on the LHS.

    The blue tape, less k rows from row 0 on to k-1, is still endless beyond k and k can be relabelled as 0 freely and set in perfect 1:1 correspondence to the un-pulled, untrimmed pink tape.

    Endlessness is decisive and points to a zone of endlessly remote ordinals.

    So, however we may group or label, such endless continuation is there on the LHS. Setting up w as assigned first transfinite does not change that.

    Thus the concerns I have expressed, which seem to me to boil down to turning the natural succession of counting sets into in effect a finite. If ALL successive counting sets are finite, are finitely remote on our punched tape models, how can there be endlessness?

    I find, to my mind something suspiciously like the race loop between unstable opposed states in the statement: this statement is false. If true, it must be false, if false, it must be true, it is self referentially incoherent.

    What has seemed so far reasonable to me, is to argue that we identify the potentially infinite and point across an ellipsis of endlessness. Then, accept that mathematical induction in the ordinary sense is of this character and can only, strictly, apply to a finite chain of values. For any value we can reach or specify that can be exceeded by another finite — note the implied do forever that cannot be completed in actuality — the general case will apply. But when endlessness is brought to bear, it must be reckoned with in its own right.

    As to speaking of a far zone, yes the term is fuzzy; as fuzzy as endlessness is and as fuzzy as an ellipsis of endlessness is. Let me spell it out as the zone denoted by endless continuation of the potentially infinite.

    In the case of the one sided endlessness of the punched paper tapes, if they are endless then as there is for all spans of the tapes a pattern of rows every 0.1 inches, there will be endlessly remote rows. And that applies for all cases to a potentially infinite succession idealised as complete.

    However we manage to reckon with it, endlessness must be taken seriously and must not in effect be reduced to finitude.

    KF

    PS: DS, you are reverting to things that were already correctively addressed above. I have no time just now to go into do forever loops.

  421. PPS: For convenience, I use Wiki on Peano:

    >>1] 0 is a natural number.

    6] For every natural number n, S(n) is a natural number. [S being, successor]

    . . . because[citation needed] 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1 and 6 define a unary representation of the natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)) (which is also S(1)), and, in general, any natural number n as the result of n-fold application of S to 0, denoted as Sn(0). The next two axioms define the properties of this representation.

    7] For all natural numbers m and n, m = n if and only if S(m) = S(n). That is, S is an injection.

    8] For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

    Axioms 1, 6, 7 and 8 imply that the set of natural numbers contains the distinct elements 0, S(0), S(S(0)), and furthermore that {0, S(0), S(S(0)), . . . } [are subsets up to] N.[citation needed] This shows that the set of natural numbers is infinite. However, to show that N = {0, S(0), S(S(0)), . . . }, it must be shown that N [are subsets up to] {0, S(0), S(S(0)), …}; i.e., it must be shown that every natural number is included in {0, S(0), S(S(0)), . . . }. To do this however requires an additional axiom, which is sometimes called the axiom of induction. This axiom provides a method for reasoning about the set of all natural numbers.[citation needed]

    9] If K is a set such that:
    0 is in K, and
    for every natural number n, if n is in K, then S(n) is in K,
    then K contains every natural number.>>

    Once successive patterns and stepwise progress are introduces as seen, the do forever loop, potential as opposed to completed infinite, ellipsis of endlessness and pointing across the loop are necessarily present also.

    And, pervade the onward work that builds on such.

    Where of course post Godel,an entity complex enough to enfold arithmetic will not be complete and coherent and we cannot constructively show it coherent though limited.

  422. PPPS: And again, if the tape is endless, it is endlessly beyond any arbitrarily large k we can succeed by k+1 etc. And if it has rows all along it by definition, some will be endlessly remote. Or else endless does not in reality mean endless and the proper subset in correspondence with the original set approach, fails. Can the blue tape be pulled in from r_0 to r-k-1 and treated as though k, k+1 on were 0, 1, on, still matching 1:1 with the pink? If yes, endlessness obtains. If not, endlessness does not mean endlessness.

  423. PPPPS: Aleta, I would suggest that there are many applications of logic to quantities and structures above, which is the essence of mathematics. Further, though simple, the application of endlessness and the pointing out of limitations on do forever chaining do have some relevant force. Where the algorithm presented at 217 above coming on two weeks past shows the main concern adequately; and highlights the effective embedding of a copy of the set of successive counting sets as a whole in its LHS listing in succession, i.e. if the set as a whole that collects successive counting sets grows to endlessness perforce so will its members in the upper zone represented by ellipsis of endlessness. I have no interest in playing out schoolbook exercises that readily go off on endless tangents.

  424. F/N: Wolfram on transfinite induction, for anchoring reference:

    >>Transfinite Induction

    Transfinite induction, like regular induction, is used to show a property P(n) holds for all numbers n. The essential difference is that regular induction is restricted to the natural numbers Z^*, which are precisely the finite ordinal numbers. [–> notice, tantamount to: the chaining in ordinary induction is inherently finite] The normal inductive step of deriving P(n+1) from P(n) can fail due to limit ordinals. [–> which have no specific, particular predecessor, due to ellipsis of endlessness, e.g. w]

    Let A be a well ordered set and let P(x) be a proposition with domain A. A proof by transfinite induction uses the following steps (Gleason 1991, Hajnal 1999):

    1. Demonstrate P(0) is true.

    2. Assume P(b) is true for all b less than a.

    3. Prove P(a), using the assumption in (2). [–> that is, chain the implication, where 2 gets around the lack of specific predecessor by saying for all b less than a ]

    4. Then P(a) is true for all a in A.

    To prove various results in point-set topology, Cantor developed the first transfinite induction methods in the 1880s. Zermelo (1904) extended Cantor’s method with a “proof that every set can be well-ordered,” which became the axiom of choice or Zorn’s Lemma (Johnstone 1987). Transfinite induction and Zorn’s lemma are often used interchangeably (Reid 1995), or are strongly linked (Beachy 1999). Hausdorff (1906) was the first to explicitly name transfinite induction (Grattan-Guinness 2001). >>

    I have commented on points. KF

  425. KF,

    Would you please answer the question I posed in #419?

    I will say, as I did previously, that the Turing Machine tape is not endless by the definition you are using if that means some of its cells must be infinitely many steps from cell 0.

  426. DS, if the tape is endless does that not mean that it is endless beyond any given k, no matter how large, where k is followed by k+1 etc? If not, then what specifically does endlessness mean? KF

    PS: Your question in its various forms has been answered any number of times. Endless means endless, just as the just above discusses and shows in the form of defining infinity on 1:1 match of a set and a proper subset, here from k on..

  427. KF,

    DS, if the tape is endless does that not mean that it is endless beyond any given k, no matter how large, where k is followed by k+1 etc? If not, then what specifically does endlessness mean? KF

    I don’t think we have a proper mathematical definition of “endless” yet. [Edit] I believe you introduced the term, so it’s your job to give a rigorous definition (preferably in terms of the Turning Machine tape).

    Do either of statements 1 or 2 in my post #419 capture the meaning?

    PS: Your question in its various forms has been answered any number of times. Endless means endless, just as the just above discusses and shows in the form of defining infinity on 1:1 match of a set and a proper subset, here from k on..

    [Edit]

    Is statement 1 or 2 correct, in your view?

    Just say “1 is correct”, “2 is correct”, or perhaps “neither is correct”, whatever the case may be.

    Saying “endless means endless” is not helpful without a clear definition to begin with.

  428. DS, First the term is not mine, it is longstanding — cf the ubiquitous ellipsis; it is time to lose the rhetorical hints of dismissive idiosyncrasy. Second so far as I can see, saying that for any given k, followed by k+1 etc, the sequence from k on can be put in 1:1 correspondence with that from 0,1, on is a definition of endlessness, which means there are legitimate values endlessly — adverb — beyond any given k that is finite. For if there were a finite run length taking away k would leave some of the original unmatched. So thirdly, I have repeatedly defined what I understand by endless, as based on a standard definition of the transfinite since Cantor. When I say endlessness is endlessness (and not endedness), I am in effect appealing to the three classic laws of thought, with A is A leading. Once world W = {A|~A} then A = A and A != ~A with A x-or A, will obtain. KF

  429. KF,

    Second so far as I can see, saying that for any given k, followed by k+1 etc, the sequence from k on can be put in 1:1 correspondence with that from 0,1, on is a definition of endlessness.

    Ok, thanks.

    But that definition of an endless Turing Machine tape is consistent with both of my statements 1 and 2, so it doesn’t tell me which you hold to.

    So again, which is correct, 1 or 2 (or neither)?

  430. kf writes,

    a definition of endlessness, which means there are legitimate values endlessly — adverb — beyond any given k that is finite.

    Yes, Dave, kf’s statement still doesn’t distinguish between your two statements, because it doesn’t clearly state what “legitimate” values mean.

    Yes, there are always an endless number of further finite values beyond any given k, and surely such values are “legitimate.” That is what your statement 1 says.

    Statement 2, however, says that there is at least one “legitimate” value that is greater than any possible k. Does kf believe this is the case? That is the question.

    kf, in the interest of mathematical clarity, should respond to your questions with yes or no answers so we can know what he means by “legitimate”. Your questions are stated with mathematical clarity, not with words whose meanings are not necessarily clear, so addressing them would help us understand what his words mean.

  431. DS, endlessness is not a synonym for finiteness. There is where my concerns lie, again. KF

  432. KF,

    DS, endlessness is not a synonym for finiteness.

    I agree.

    Which of my statements from post #419 are true of the infinite Turing Machine tape, if any?

    Why won’t you answer this simple question?

  433. Aleta, again and again, if something is endlessly beyond ANY finite value k then there is a problem with an argument that entails that all counting sets denoted as natural numbers — that is what legitimate counting numbers are — will be finite but there is an infinite supply of same; esp when that argument rests on ordinary mathematical induction which is closely tied to finite stepwise succession. At minimum, paradox. Or else, use of finite and infinite in ways that run perilously close to infinite and finite are two ways of saying the same, rather than that they denote alternatives . . . something suspiciously close to A AND ~A, thence ex falso quodlibet. In terms of the paper tape, if there is a 0.1 inch pitch, then for any finite value k, the distance from the near end will be k * 0.1 in inches. A finite value exceeded by the k + 1th row. If all possible k are finite, the tape will be finitely long not endless as length to k is k* 0.1. Maybe something so concrete as this may help clarify the concerns to you. KF

  434. DS, see the just above, KF

  435. KF,

    I’m about 80% confident that you believe statement 2 is correct (and necessarily 1 as well, it being strictly weaker than 2).

    Am I correct?

    It would be helpful to us in understanding your position if you would simply say “yes” or “no”.

  436. Reminder: “2) There exists a cell C such that for every natural number n, cell C is beyond cell n”

    True or false?

  437. DS,

    we go in circles, needlessly.

    As I have already had occasion to point out, the inquisitorial y/n demand when there is an obvious conceptual difference in play is worse than useless.

    Again, the pivotal concern is what is endlessness, and the pink and blue punch tapes make it concrete.

    Their successive rows correspond to natural numbers, starting from 0 at the near end. For any — repeat, any — finite row k, a further row k+1 will exist as a bound that is succeeded by k+2 etc. For the blue tape to be endless, truncating it at k and putting k, k+1 etc in match with the pink one at 0, 1, etc will still preserve a 1:1 match.

    If they are not endless, then we will not have such a match of proper subset with the original set.

    As a direct consequence, for ANY finite k, there will be an endless onward tape k + 1, k+2 etc — something you saw quite well when you argued that at any given finitely remote past time there were onward values of time on an infinite past view.

    If there is such an endless — infinite — onward run of tape for any particular finite k we please, then the span of the tape beyond must recede endlessly, there is no upper finite limit to its length.

    As rows exist every 1/10 inch, there will be endlessly remote onward rows that no finite, stepwise count process can span. It is reasonable to say those rows are infinitely beyond.

    Now, we may wish to proceed thusly, and define a set A that takes in all we can potentially count to stepwise in finite stages, caveat being that ability to exceed any k implies all in A are finite.

    We can then identify A as indefinitely large or at least extensible. Potentially infinite. But to go on to the ideal extension of that potential by closing off the set is to point across an ellipsis of endlessness.

    Is A another label for N, the set of successive counting sets as labelled?

    The argument put on the table answers yes, per how ordinary mathematical induction is applied.

    I have concerns on doing this, as N is defined to include the ellipsis of endlessness:

    {0,1,2 . . . k, k+1, . . . }

    Indeed, the way w succeeds is such that it is said not to have a specific, finite predecessor, z –> w, z being finite and the last natural number. That would have opposite effect to the intent. Instead w succeeds the endlessness and represents order type of the whole.

    The ellipsis of endlessness comes into play crucially.

    And in my mind if it is to be taken seriously it involves endlessness in the set N, which is beyond any count zone A reachable in finite steps.

    Hence my concerns tied to the concept of an infinite succession of finite numbers. For as the counting sets mount up to endlessness, that points to endlessness in successive sets, making them in effect tend towards being copies of the whole.

    KF

  438. Aleta, As just noted to DS, the issue pivots on for any FINITE k, there will be onward cells or rows k+1 etc. The point in doubt is whether N — said to be endless — is exhausted by any large enough k. I hold, not on grounds that once k is finite we have k+1 etc. So, to use n instead of pointedly finite k is problematic. KF

  439. KF,

    Their successive rows correspond to natural numbers, starting from 0 at the near end. For any finite row k, a further row k+1 will exist as a bound that is succeeded by k+2 etc. For the blue tape to be endless, truncating it at k and putting k, k+1 etc in match with the pink one at 0, 1, etc will still preserve a 1:1 match.

    If they are not endless, then we will not have such a match of proper subset with the original set.

    As a direct consequence, for ANY finite k, there will be an endless onward tape — something you saw quite well when you argued that at any given finitely remote past time there were onward values of time on an infinite past view.

    If there is such an endless — infinite — onward run of tape for any particular finite k we please, then the span of the tape beyond must recede endlessly, there is no upper finite limit to its length.

    Yes, I agree. (Notice how I am willing to precisely state my position so as to facilitate our communication.)

    As rows exist every 1/10 inch, there will be endlessly remote onward rows that no finite, stepwise count process can span. It is reasonable to say those rows are infinitely beyond.

    No, that is not a reasonable thing to say. This is in essence equivalent to my statement 2. You are saying that there exists at least one row/cell such that this cell is beyond cell n, for all finite natural numbers, which is not the case.

    Now, we may wish to proceed thusly, and define a set A that takes in all we can potentially count to stepwise in finite stages, caveat being that ability to exceed any k implies all in A are finite.

    Is A another label for N, the set of successive counting sets as labelled?

    Yes, I believe that is correct.

    Indeed, the way w succeeds is such that it is said not to have a specific, finite predecessor, z –> w, z being finite and the last natural number. That would have opposite effect to the intent. Instead w succeeds the endlessness and represents order type of the whole.

    Ok, if you’re saying that ω is greater than any natural number, yes, that’s true.

    And in my mind if it is to be taken seriously it involves endlessness in the set N, which is beyond any count zone A reachable in finite steps.

    The set is endless according to your definition above (in fact it coincides with the definition of “infinite”).

    How that leads to some zone unreachable in finite steps, I have no idea. In fact I’m certain it doesn’t.

    Consider the infinite Turing Machine tape. There would have to be a leftmost cell not reachable in finitely many steps from cell 0. Then the cell immediately to its left would be reachable in finitely many steps from cell 0.

    Do you think that is possible?

  440. KF,

    Incidentally, have you studied formal languages? For example, the set of all bit strings, each of which is a finite sequence of 0’s and 1’s. Examples: 0, 101001, and 1111.

    Each bit string has finite length, but there are infinitely many of them. Erm, right?

    In fact, each bit string (except for the empty string) corresponds to a (finite) natural number by considering it as a binary numeral. So we have an obvious mapping from the set of these bit strings to N, which is onto (or surjective), which means the set is at least as large as N. (In fact, the set of these bit strings has the same cardinality as N). [Edit: Likewise, any language over an at most countable alphabet is also countably infinite].

    This is just one more of a huge number of examples of infinite sets, each element of which is finite in some sense. These things are common as dirt in mathematics.

  441. DS, in a for a moment brief, again, finite and infinite are not synonyms; if the tapes by their nature are punched at every 1/10 in in rows, and they run on endlessly, by logic they will have endlessly remote rows; where as was already highlighted endlessness will mean that from any finitely remote row k from the origin at the near end of the tapes, there will be yet endless onward rows. That is how the blue tape can be pulled in k rows and rows k, k+1 etc set in 1:1 correspondence with row 0,1 etc of the pink tape. I note also that for a given string length n, the number of possible bit combinations is 2^n which for finite n will be finite; from which we can easily infer that the total of possible strings of length 0 up to length k will be finite, and k+1 will also be finite and so forth. For some k, we simply count up from 000 . . . 0 [k digits] to 111 . . . 1. If you remove the finitude of n, you will imply that there are endlessly many strings of up to endless length, but that is not the point you were trying to make and it was never in dispute. KF

  442. kf writes,

    endlessness will mean that from any finitely remote row k from the origin at the near end of the tapes, there will be yet endless onward rows

    We all agree with this. This is equivalent to Dave’s statement 1 at 419.

    1) For every natural number n, there exists a cell C beyond cell n.

    Since this is true, no matter what finite number n you are at, there is still an endless progression ahead of you.

    There is no end to the natural numbers. Therefore, there is no largest natural number. Therefore, there are an infinite number of natural numbers.

    Don’t we all agree with these things?

    What do we disagree about?

  443. KF,

    DS, in a for a moment brief, again, finite and infinite are not synonyms;

    Of course not. The set {1, 2, 3} cannot be put into 1-1 correspondence with any of its proper subsets. N can. {1, 2, 3} is a finite set, while N is not.

    if the tapes by their nature are punched at every 1/10 in in rows, and they run on endlessly, by logic they will have endlessly remote rows;

    You certainly haven’t demonstrated the existence of any of these “endlessly remote” rows. There are cells arbitrarily many steps from cell 0, but they are all finitely distant.

    As to it being a matter of logic, you are quite literally the only person I have ever heard/read make such claims, that I recall anyway. Have you noticed that no one here has weighed in on your side on this matter? I defy you to find any support for your position anywhere on the internet outside of 4chan.

    All you have shown with you blue/pink tape illustration is that the cells on one tape can be put into 1-1 correspondence with the cells numbered k and up on the other tape, for any finite k you please. That just means the tape is infinite.

    I note also that for a given string length n, the number of possible bit combinations is 2^n which for finite n will be finite; from which we can easily infer that the total of possible strings of length 0 up to length k will be finite, and k+1 will also be finite and so forth. If you remove the finitude of n, you will imply that there are endlessly many strings of up to endless length, but that is not the point you were trying to make and it was never in dispute.

    Again, the way you’ve phrased this leaves me with questions. “Of up to endless length”? I don’t know what that means. So:

    Do you agree that the set of all bit strings of finite length has cardinality aleph-null?

    To be clear, I’m referring to the language normally denoted {0, 1}^*, and there are no “infinite” bit strings included.

  444. DS, if there are endlessly many rows at 0.1 inch per row, they are endlessly remote in distance, a proxy for scale. KF

  445. So, you’re saying that there are rows that one could never get no matter how long one let the tape run. True?

  446. KF,

    DS, if there are endlessly many rows at 0.1 inch per row, they are endlessly remote in distance, a proxy for scale. KF

    It is not the case that there exists a particular row such that for all natural numbers n, this row is greater than n inches from row 0.

    It is the case that for every natural number n, there exists a particular row which is more than n inches from row 0.

    Right?

  447. DS, the point is, that for any finite step by step procedure — and as even place value notation is a power series in disguise that is caught up — there will be endlessly more rows beyond it, so that there is what I have called a far zone beyond the reach of any finitely bound procedure; often indicated by an ellipsis of endlessness . . . which is doing a lot of often unrecognaised heavy lifting. That is what I have been pointing to as a fundamental phenomenon of endlessness. Analysis of systems and structures with endlessness in them must address that. This includes that something is inherently limited in ordinary mathematical induction, and when one jumps to transfinite, it is by no means clear that set A which is reachable by finite processes — cf Wolfram as cited above — is capturing the full set of successively larger counting sets aka counting numbers and/or that of the relevant ordinals. KF

  448. KF,

    Well, I can’t force you to answer this key question.

    Anyway, to support your position (as I understand it), here’s what you need to do:

    Start with the pink and blue tapes having their cells 0 aligned initially on the left. Both tapes extend infinitely far to the right.

    Consider finite leftward shifts of the blue tape by k cells, where k is a natural number, so that cell k of the blue tape ends up next to cell 0 of the pink tape.

    You need to show that there is a cell C in the blue tape which will never end up next to or to the left of cell 0 of the pink tape, regardless of which k was chosen.

    That’s going to be impossible in view of the fact that the cells of each tape are in 1-1 correspondence with the natural numbers, but let us know what you come up with.

  449. Addendum to the last sentence of my post #450:

    That’s going to be impossible in view of the facts that the cells of each tape are in 1-1 correspondence with the natural numbers, and cell k + 1 is adjacent to cell k, for each natural number k.

  450. DS, I have already repeatedly pointed out the significance of endlessness, which means that there will be any number of rows in the far right zone (8 bits per row in 5+3 tape . . . ) which will be beyond any finite value k.

    Let me “sketch” the blue, with the cutoff:

    0 ===//===| k ===> . . . EoE . . .

    Endlessness means just what it says and is the operational sense of infinite. That seems to be close to the core issue: infinite is not a synonym for finite. So when — as repeatedly pointed out — the blue is pulled in k rows, 0 to k – 1, from k on there will be an endless 1:1 correspondence with the pink tape, k -> 0, k+1 -> 1, etc. And, the pull-in can then be repeated any number of times you please, to the same effect, rather like the put existing guests in even rooms and put endless guests in the odd ones with Hilbert’s hotel. That is bound up in the meaning of endlessness. And, that this point is so hard to see underscores the conceptual issue that makes a simplistic yes/no answer meaningless. Endless means endless, and the pull in k and align in 1:1 correspondence shoes the remaining onward rows — punched with sprocket holes and up to 3 _ 5 holes at 1/10 in pitch all the way — will be an infinite, ordered set. Which means, due to endlessness, there will always be onward endlessly many values than any finite k. Not just one more, endlessness is pivotal. Operationally, that is set up by the 1:1 match after pulling in. KF

  451. #452 KF

    Which means, due to endlessness, there will always be onward endlessly many values than any finite k. Not just one more, endlessness is pivotal.

    Each of those values though is finite. Counting up one at time never gets you to an infinite value in a finite number of iterations. You don’t all of sudden traipse into the ‘transfinite’. You will eventually get to any specified value and you should even be able to predict when you get to it. But you won’t get to ‘infinity’.

  452. KF,

    DS, I have already repeatedly pointed out the significance of endlessness, which means that there will be any number of rows in the far right zone (8 bits per row in 5+3 tape . . . ) which will be beyond any finite value k.

    For any natural number k, there are infinitely many rows in the tape beyond row k.

    If that’s what you’re saying, yes.

    There are no rows in the tape beyond every row k.

    If you’re asserting that there are, no.

    If you want to have an actual discussion about this, then you’re going to have to take a clear position on this issue.

    Endlessness means just what it says and is the operational sense of infinite.

    So why not just say “infinite”? The tapes are infinite because the cells can be put into 1-1 correspondence with (some) proper subsets of the cells.

    That seems to be close to the core issue: infinite is not a synonym for finite. So when — as repeatedly pointed out — the blue is pulled in k rows, 0 to k – 1, from k on there will be an endless 1:1 correspondence with the pink tape, k -> 0, k+1 -> 1, etc.

    And I have repeatedly agreed with this.

    And, the pull-in can then be repeated any number of times you please, to the same effect, rather like the put existing guests in even rooms and put endless guests in the odd ones with Hilbert’s hotel.

    Yes, repeatedly agreed to already.

    And, that this point is so hard to see underscores the conceptual issue that makes a simplistic yes/no answer meaningless.

    I hope this trend doesn’t catch on at UD. It gets criticized quite a bit, but I think most people here accept the responsibility to respond to questions in a debate. Especially one tagged as “Darwinist rhetorical tactics”.

    There is no reason you can’t answer my post #419.

    Endless means endless, and the pull in k and align in 1:1 correspondence shoes the remaining onward rows — punched with sprocket holes and up to 3 _ 5 holes at 1/10 in pitch all the way — will be an infinite, ordered set.

    Yes, yes, agreed to once again.

    Which means, due to endlessness, there will always be onward endlessly many values than any finite k.

    For any particular natural number k. There are no particular values beyond every natural number k.

    ***

    So, your post should end with something like, “therefore cell C in the blue tape will never end up next to or to the left of cell 0 of the pink tape, regardless of the value of the natural number k”.

    If you have to resort to a nonconstructive proof, that’s ok with me also.

  453. KF,

    PS to my post #454:

    You spend quite a bit of time here speaking of the importance of logic, the rules of right reason, and so on.

    I also think it’s important to be clear about precise logical structure of statements one makes.

    All Aleta and I are asking you to do in #419 and following is to tell us which of these (if either) statements you are making:

    1) ∀ n ∃ C P(C, n)

    2) ∃ C ∀ n P(C, n)

    where P(C, n) means “cell C is beyond cell n”.

    In other words, to translate your own statement into one of these forms (or perhaps something else, if necessary).

  454. EZ, the problem is endlessness, which is flat contrary to finite. KF

  455. DS,

    I have already stated and showed that on endlessness, for any finite K, there will be endlessly many more rows than the row k, onwards. I have now even illustrated:

    0 ===//===| k ===> . . . EoE . . .

    Where the blue tape from k on can be matched 1:1 with the pink one from 0, precisely because of endlessness:

    SNIP, MATCH:

    | k ===> . . . EoE . . .

    | 0 ===> . . . EoE . . .

    Where, once a subset can be matched to the full set, the original set is transfinite.

    Indeed, by going out k’, we can cut and match again showing k on is transfinite.

    And so on arbitrarily many times.

    Where per composition of the tape the rows are loaded with holes, at 0.1 in pitch.

    So, in the far zone included by the ellipsis of endlessness, there will be holed rows, endlessly remote from 0 and k etc.

    By the logic of the tape’s composition and endlessness.

    Which is an ordered, counting succession.

    This means, the issue raises points on how we describe and define our sets. The endlessness is primary, the algebra and definitions etc around it are secondary.

    So I have reason to be concerned, including when ordinary mathematical induction is brought in, as has already been highlighted. Even, when transfinite induction is brought in — as was briefly noted on relation between A and the ordered sequence of counting sets.

    If that does not suffice to focus the concern, then something is seriously wrong with your argument.

    KF

  456. KF,

    I have already stated and showed that on endlessness, for any finite K, there will be endlessly many more rows than the row k, onwards. I have now even illustrated:

    Yes. That corresponds to statement 1 in my post #455.

    So, in the far zone included by the ellipsis of endlessness, there will be holed rows, endlessly remote from 0 and k etc.

    Now this is ambiguous again, depending on what “endlessly remote” means.

    If by this you mean “at arbitrarily large finite distances” yes. That is consistent with the first quote I clipped, and is statement 1 from #455.

    If you mean “at infinite distance”, no. That’s statement 2 from #455.

    Am I right in assuming you mean “at arbitrarily large finite distances”?

  457. #456 KF

    EZ, the problem is endlessness, which is flat contrary to finite. KF

    What is the different between ‘endlessness’ and ‘infinity’? What is your mathematical definition of ‘endlessness’? Why is ‘endlessness’ a problem? Since Cantor much work has been done dealing with these issues and the vast majority of mathematicians have come to accept the Cantor view; how does your approach differ from his?

    Just out of curiosity, what mathematical courses have you taken? Have you run your ideas past some other mathematicians? Like all sciences mathematics is tricky and complicated and even those who are good at it make sure their ideas make sense by checking with colleagues and fellow researchers.

    Again, if you initiate a counting procedure you cross many finite steps but you never get to the transfinite/infinite. There is no boundary or edge. And this situation has been researched and dealt with.

  458. EZ, I have already cited, but will repeat for convenience, here AmHD:

    in·fi·nite (?n?f?-n?t)
    adj.
    1. Having no boundaries or limits; impossible to measure or calculate. See Synonyms at incalculable.
    2. Immeasurably great or large; boundless: infinite patience; a discovery of infinite importance.
    3. Mathematics
    a. Existing beyond or being greater than any arbitrarily large value.
    b. Unlimited in spatial extent: a line of infinite length.
    c. Of or relating to a set capable of being put into one-to-one correspondence with a proper subset of itself.

    n.
    Something infinite.
    [Middle English infinit, from Old French, from Latin ?nf?n?tus : in-, not; see in-1 + f?n?tus, finite, from past participle of f?n?re, to limit; see finite.]
    in?fi·nite·ly adv.
    in?fi·nite·ness n.

    KF

    PS: Above that property of being matched with a proper subset 1:1 is indicated, and it shows how infiniteness and endlessness beyond any arbitrarily large but finite value are inextricably interconnected.

  459. #460 KF

    Yes, I know what the definition of infinity is (please don’t be patronising) and I know how mathematicians found a way to deal with it. But you’re not specifying how you discern ‘endlessness’ from infinity. And you’ve not indicated what your mathematical background is. And you’ve not (apparently) put your ideas forward for review by other mathematicians.

    We’re not some novices that are just going to accept your statements without question if we don’t understand how you’re using terms. Science and mathematics progress via a process of proposals, scrutiny, revision, acceptance, repeat. What scrutiny have you put your proposals to?

    Above that property of being matched with a proper subset 1:1 is indicated, and it shows how infiniteness and endlessness beyond any arbitrarily large but finite value are inextricably interconnected.

    So, endlessness and infinity are the same to you? But then how do you rectify the idea of an ENDLESS sequence converging to a finite value? This is why we’re trying to figure out how you’re using the terms. For example:

    1.1, 1.01, 1.001, 1.0001, 1.00001 . . . is an endless sequence which approaches a finite value. At no point do the terms of the sequence achieve a transfinite value. We might say the limit of a-sub-n (a-sub-n being the nth term) as n goes to infinity is 1. I think this example is pretty clear.

    But . . . likewise . . .

    1, 2, 3, 4 . . . is an endless sequence. It diverges to infinity meaning the individual terms will eventually ‘beat’ any limit you specify. But, again, each term in the sequence is finite. And we would say: the limit of a-sub-n = infinity even though no term is infinite.

    We can work with infinite series instead of sequences if you like. Part of my reason for asking after your mathematical background was so I could pick more meaningful examples.

    A couple of infinite series examples:

    1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + . . . . diverges (becomes infinite, does not ‘level off’ to a finite value, will continue to grow without bound EVEN THOUGH the individual terms converge to zero) but

    1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . converges (approaches a finite value, gets closer and closer to 2 (from below) the more terms you add on). It never exceeds 2 even though you can add ‘endless’ terms together.

    All of this is standard, first year Calculus stuff. I learned it when i was 18 years old. And it works. It’s useable. Just look at Fourier transforms (infinite series), something used by engineers every day. There’s no great mystery or magic about it. Non-controversial, bread-and-butter, applied mathematics. And there’s infinities and ‘endless sequences and series everywhere. With no problems.

  460. This thread appears to be endless, but unfortunately not because it is progressing. (Although I welcome ellazimm and his/her? succinct and clear points.) Rather then k, k + 1, k + 2, … we appear to have k, k + 1, k, k + 1, …

    Therefore a summary might be in order, in order to bring the discussion to an end. There are two areas to summarize:

    A. the mathematical issues themselves, for which we need to distinguish the statements which all parties agree upon from those that there is either clearcut disagreement, or, more commonly, confusion about what exactly is the issue at hand.

    B. the features of the discussion that are causing it to be stuck in an endless loop rather than making progress.

    So here we go. First, some of the reasons we aren’t getting anyplace.

    B1.1 I would like to indicate points we agree on, and points where there is disagreement. However, this is hard to do because we can’t figure out exactly and specifically exactly what point kf is making. Some reasons:

    B1.2 kf uses a number of informal and undefined terms, such as “past the ellipsis”, “far or remote zone”, “infinitely or endlessly remote”, “heavy lifting”, etc. Since he doesn’t explicitly define these, either in words and especially not in mathematical language, we don’t exactly know what he means. Trying to become clear on exactly what specific mathematical point he objects to and/or is offering has been one of the main themes of this discussion

    B1.3 But when asked to answer specific questions about what he means, or if he agrees or disagrees with specific mathematical statements, he refuses, calling such “inquisitorial y/n” questions.

    In fact, he says, “The endlessness is primary, the algebra and definitions etc around it are secondary. That is, he continually repeats his vague and unspecified “concerns” about endlessness, but dismisses attempts to define terms and make specific mathematical claims as “secondary.”

    B1.4 kf continually repeats points that we have agreed upon with many times, as if they were points of contention. He doesn’t seem to be able to remember and/or acknowledge such points (see part A for specifics). Among other things, this adds to the length and repetitiveness of his posts, and contributes to the sense of not being able to figure out exactly what he disagrees with us on.

    Mathematical issues

    A1.1 The natural numbers start with zero, and are such that for every natural number k, k + 1 is also a natural number. Dave as stated this in formal language as “For every natural number n, there exists a number [cell C] beyond n (using kf’s concrete example of a tape to represent stepping through the natural numbers.)

    A1.2 This definition ensures that all are natural numbers are finite.

    I’m sure we all agree about A1.1 and A1.2

    A1.3 The set of natural numbers is infinite.

    I’m pretty sure that kf agrees with this for two reasons.

    A1.3a He often refers to the fact that the natural numbers N can be put in a 1:1 correspondence with a proper subset of itself

    A1.3b He often references w and aleph null as transfinite numbers which represent the order of infinity associated with the natural numbers, and that w > the ordinal for any natural number. He accepts that N is an infinite set.

    A1.4 However, kf seems to object to the statement “There are an infinite number of natural numbers”, although that seems to me to be exactly what A1.3 says.

    What exactly does kf seems to believe? He seems to believe that there is a “far or remote zone” “past the ellipsis” that contains “cells” that are “remotely distant from zero”.

    That is, he seems to think that there is a part of the set of natural numbers that is forever beyond the reach of the finite natural numbers. Dave has expressed this as, “There exists a cell C such that for every natural number n, cell C is beyond cell n.”

    Dave has asked kf to state whether this represents his position, but kf refuses to answer, seeing it as one those inquistorial “gotcha” questions. 🙂 But as far as I can tell, this is what kf believes.

    But this makes no sense to us It seems to say that there are numbers in the set of natural numbers that are not natural numbers, although it is not clear whether kf’s “far zone” actually has numbers in it. It’s just not clear what he thinks, frankly.

    But, in conclusion, given kf’s unwillingness and/or lack of ability to be mathematically specific, in conjunction with the other points listed in section B above, I don’t believe there is any chance of us getting any more clarity and specificity on the subject that we have gotten.

  461. That’s a nice summary, Aleta.

    This might be wishful thinking, but I’m holding out hope that the endless loop of this thread might turn out to be an endless spiral, where even though we keep cycling through the exact same questions over and over, we get a little closer to the truth and commit fewer errors each time.

  462. Optimism is good. 🙂

    One of the reasons I participate in discussions like this is for the exercise of honing the clarity and specificity of my own understanding, and from learning from others who perhaps believe as I do but have different ways of expressing themself, or who bring up points that I haven’t thought of or didn’t know about. From that point of view, this has been an instructive thread.

  463. It could be that I’m too cynical for the web, but to me it is exceedingly clear why KF resorts to such vague language and why he refuses to answer clear yes/no questions.

    Were he clear, it would be apparent to many, maybe even to him, that he is wrong. However, this can’t be since: “KF is right and DS is wrong. And if math agrees with DS, then math is wrong, too.”

  464. EZ & Aleta:

    (& DS, attn HRUN . . . who needs to learn what has been pointed out several times but studiously ignored: that Mathematics sometimes advances by challenging the consensus so appeal to authority rather than the logic of structure and quantity, fails . . .),

    It seems I need to explain my perspective, providing my own summary of why I have concerns with the way the transfinite [Cantor’s term of choice] has been discussed and operationally used.

    As a preliminary, please read 215 – 217 above . . . esp 217, and then . . .

    First, it is clear that this thread has long since established the main point in the OP, that there is a major problem in positing an endless past leading up to the present.

    Traversing the endless and completing such a traverse in finite stage steps is futile.

    Appealing to what is tantamount to it, is a fallacy.

    That’s Spitzer’s point in a nutshell.

    He is right, to end the endless is a self contradiction on the level of a square circle. The criteria for the one cannot be met while meeting the criteria for the other.

    And, the pink vs blue punch paper tape examples with rows of 5 +3 bit cells at 0.1 inch pitch make this issue concrete.

    Once the number of rows is endless, being punched into the tape all along its run, we can never advance to traverse the endless in finite stages. Run along the sprocket-holes sufficiently to get to ANY finite k, however large, and we face the problem that thereafter — exactly because the tape runs on endlessly to the right hand side for convenience — rows k, k+1 etc can be put into 1:1 correspondence with the un-moved but equally endless pink tape from rows 0, 1 etc.

    Endlessness is pivotal, and it provides an operational definition of what it means for the ordinal sequence of counting sets or numbers to run on without end and be transfinite.

    Let me illustrate for tapes:

    |0, 1, 2 === . . . k, k+1 [finite values] . . . ===> . . .

    |k, k+1 [finite values] . . . ===> . . . (matching 1:1)

    as well as for the set:

    {0,1, 2 . . . k, k+1, . . . }

    with w etc as successor in the transfinite zone:

    {0,1, 2 . . . k, k+1, . . . } –> w, w+1, w+2 . . . w+g . . . [ mount on up to epsilon-zero etc]

    [I quietly note on the surprise well above when the legitimacy of that continuation was pointed out:

    0,1, 2 . . . k, k+1, . . . [Ellipsis of endlessness] . . . w, w+1, w+2 . . . w+g . . . [ mount on up to epsilon-zero etc]

    where the interval (0,1) — open — can for contemplation be catapulted in to fill in between w and w+1 etc using mild enough infinitesimals and the 1/x hyp function, at least as an exploratory model. This seems to allow unification of the transfinite zone, at least as a suggestion to be looked at.

    To see what this implies for descending from an endless past, simply reverse the tape. By logic, the same span now runs off to the LHS, and if the span cannot be traversed in 0.1 inch steps one way, it cannot be traversed the other way either.

    There is a major problem with worldviews that either have to pull a world out of non-being or else have to pull a world out of an endless, transfinite past to get to the present.

    In the course of such an issue being on the table, questions on the natural numbers and the claim that their span is transfinite but every particular natural number is finite and bounded came up.

    This has seemed at minimum paradoxical to me and much of the thread has circulated around this matter.

    A key issue is the nature of ordinary mathematical induction which sets up a case 0 or 1 then hangs the chaining implication C-k => C-k+1, and infers therefrom to all cases in succession. Or sometimes the all in succession is presented as simply all cases.

    What actually is entailed operationally is a do forever successively incremented loop with finite stage increments. The implication with a running range stepping k to k+1 and hanging on a first case entails that. So, at some stage we have an ellipsis of endlessness and we point across what we cannot actually span in steps.

    So, a lot hangs on the ellipsis of endlessness and the sub axiom of (often implicitly) pointing across it even though we may not span it in finite increment steps. And BTW, strictly, a sequence converging to a finite limit that is infinite is never actually completed absent “case infinity,” it just converges closer and closer so it is useful to recognise that this never actually completes an endless process either. The moreso if the trend to the infinitesimal steps is such that the relevant series diverges, i.e. the set — sequence — of successive partial sums will at some stage exceed any arbitrarily large but finite value and thereafter will be forever beyond it, or else will oscillate without converging as does the sequence [-1]^n.

    Pointing across an ellipsis of endlessness is pervasive in modern mathematical praxis.

    So, long since it has been pointed out that there is a major difference between a sequence that is convergent on going to infinitesimal increments that beyond some member will always be within a delta neighbourhood of a limit, and one that diverges by increasing in finite stages, stepwise without limit so that it goes to endlessly large values.

    Zeno’s paradoxes and kin have been off the table from the beginning.

    And it is relevant that the place value notation system is a disguised power series that goes on to endlessness, whether the focus is the fractional part or the whole number part.

    In this context, when I hear complaints oh you are vague, I see that as such is given in the face of specific cases, explanations and terms, we are dealing with conceptual gaps that reflect a break between paradigms.

    I was raised mathematically in a world of the primary infinite, where one points to the RHS or the LHS of the line of reals on a graph paper with arrow heads to indicate ranging on endlessly. I was raised on curve sketching where asymptotes approach a limit line and value endlessly but never touch. I was raised on sequences of partial sums that converge within delta neighbourhoods as they move to a limit. I was raised on differentiation from first principles and the limit approach, finding that the nonstandard analysis makes sense of infinitesimals. Except for some odd sounding claims on hyper reals and infinitesimals as in effect beyond the real range.

    In that context the infinite as what goes on endlessly is a primitive, a first point of reference. And, whole numbers, counting numbers, come up as uniformly separated mileposts on the real line, often marked in graphs with hash lines.

    We can go back and model the succession of sets per von Neumann, from {} –> 0 to {0} –> 1 etc, and go to rationals, reals and a complex plane, as models that go from [co-ordinate] geometry to algebra in effect. We may systematise and deduce, but when the algebraic transfinite clashes with the infinite in the primary sense, some warning flags will trip.

    And I see the adroitness in speaking of hyper reals as beyond reals: we have a separate model that works around the problems. Never mind, [0,1] is a CLOSED interval on the line of reals, going back to good old Allendoerfer and Oakley. So, any value, any point in the interval will be a real, filled in by using the power series endlessness of fractional place value notation.

    What we have is that in the near, infinitesimal neighbourhood of 0, there is a cloud of values that by using y = 1/x as a hyperbolic catapult, can be projected to a transfinite, hyper real zone, including mile posts at whole number values. And by simple addition, such an infinitesimal cloud can be shifted to the neighbourhood of any particular value we please.

    That is the context in which I spoke of a mild infinitesimal m being catapulted to a mile marker A in the trans finite zone across a sahara of the ellipsis of endlessness. Then, I applied a milepost by milepost down count that brings to bear the challenge of spanning the endless to reach a finitely near neighbourhood of zero. The span the endless in finite stage steps challenge kills that.

    For me at least, this gives some teeth to the claim that one cannot span an infinite actual past to arrive at the present.

    Of curse such was challenged, as all naturals are finite. That is the context in which the onward span to w, w+1 etc came in and A was identified as w + g, g a large finite value. Take it as a model that seeks at minimum to find a what if unification of what is patently a divergent cluster of models on the transfinite. Hyper reals beyond the reals, all naturals are finite but the set as a whole is transfinite and whatnot.

    The problem pivots on the case of the naturals.

    That is where pink punch paper tape vs blue comes in. By definition, endless and punched every 1/10 inch. From the near end off to an endless RHS. Think, endless cycles of 1 to 255. By the logic of endlessness, there are rows in succession all along the tape and some will be in an endlessly remote zone off to the RHS. That is, for any arbitrarily large but finite k, there will be endlessly more onward rows, receding into a zone of the endlessly remote. So, if we pull in blue k rows, k, k+1 etc can be put into 1:1 correspondence with the un-moved pink tape, endlessly. And we can do the same on a do forever loop without changing the result.

    That is the operational, primary meaning of infinite.

    On that logic, there is a transfinitely remote/distant zone of the two tapes in which that endlessness is found, an endlessness that cannot be traversed, as it has no upper END, it is utterly unbounded. Zone is vague, but it is better to be roughly right than to be exactly wrong.

    That’s Kelvin IIRC.

    As to what it means for there to be endlessly more rows beyond any finite k, that put k on in 1:1 correspondence with the pink tape suffices. And no, it is not just that k has k+1 beyond it, the endlessness represented by that innocuous seeming three dot ellipsis is pivotal.

    And, we are forever implicitly pointing across it, implying do forever loops that cannot arrive at an actual completion at the transfinite.

    So, ordinary induction shows that we can chain a conclusion in a do forever, but always finite loop. It will be reliable for any finite value, but to point across the infinite endless ellipsis and conclude all natural numbers are finite is tantamount to saying that the endless tapes are endless but have only finitely remote points. But beyond ANY finite value no matter how high, there is an endless continuation.

    That endlessness does not vanish into finitude, it is there, and by force of logic there is a remote zone beyond any arbitrarily large but finite row number. One that we can in effect secondarily model as transfinitely remote. Use a hyperbolic function wormhole to catapult from the infinitesimals near the row 0 point to leap to it if we would. As opposed to walking there in steps.

    This is the root of my concerns, and there is nothing in the thread above that would lead me to conclude that I am being hyper-concerned on something that is not significant.

    Just the opposite.

    KF

    PS: I should note that when I spoke of an implicit premise is doing heavy lifting I took time to explain. I use analogy of the modal ontological argument, in which one premise determines the outcome and stands by itself as a result. That is if an argument is of form p1, p2, p3 . . . pn + ph => c1, where c1 crucially depends on accepting ph not just the other premises, ph is the one doing the heavy lifting to reach c1. Which is a reasonable way of speaking. So the debate shifts to, why accept ph. The heavy lifter premise. This then may lead to a [quasi-] worldview level discussion on alternative clusters of premises or in effect explanatory models. In the relevant cases, we are forever pointing across acknowledged or implicit ellipses of endlessness and implying associated do forever loops of succession. These appear right there in the axiomatic framework and so it is proper to highlight the point. Indeed, this is apparent in ordinary mathematical induction and it is even implicit in the transfinite form once we see the oh the thing is so up to this threshold and from that it will follow it is true on the successor case, filling up set A. But by now I suppose I am simply speaking for record.

  465. Captcha has gone to images now!

  466. KF,

    All that typing, even including a PS on logical matters, but you still refuse to tell us which of:

    1) ∀ n ∃ C P(C, n)

    2) ∃ C ∀ n P(C, n)

    you subscribe to!

    I was going to address some of your claims in #466, but it’s so full of misconceptions, that’s an overwhelming task. Sorry KF, it really is. Maybe I’ll chime in later, but for now I’ll leave it for others to respond to.

  467. 466 KF

    This

    0,1, 2 . . . k, k+1, . . . [Ellipsis of endlessness] . . . w, w+1, w+2 . . . w+g . . . [ mount on up to epsilon-zero etc]

    is just non-sensical. You don’t cross a line and get infinite values when counting up naturals one at a time. “mount on up to epsilon-zero, etc”? What?

    Followed by

    where the interval (0,1) — open — can for contemplation be catapulted in to fill in between w and w+1 etc using mild enough infinitesimals and the 1/x hyp function, at least as an exploratory model. This seems to allow unification of the transfinite zone, at least as a suggestion to be looked at.

    makes no sense either.

    A key issue is the nature of ordinary mathematical induction which sets up a case 0 or 1 then hangs the chaining implication C-k => C-k+1, and infers therefrom to all cases in succession. Or sometimes the all in succession is presented as simply all cases.

    What actually is entailed operationally is a do forever successively incremented loop with finite stage increments. The implication with a running range stepping k to k+1 and hanging on a first case entails that. So, at some stage we have an ellipsis of endlessness and we point across what we cannot actually span in steps.

    See, you have this notion that all of a sudden you’re going to traipse into the infinite when incrementing step-by-step. And that just doesn’t happen. As you’ve been told over and over.

    In this context, when I hear complaints oh you are vague, I see that as such is given in the face of specific cases, explanations and terms, we are dealing with conceptual gaps that reflect a break between paradigms.

    It would help if you used the accepted mathematical paradigm instead of interpreting it in your own way!!

    What we have is that in the near, infinitesimal neighbourhood of 0, there is a cloud of values that by using y = 1/x as a hyperbolic catapult, can be projected to a transfinite, hyper real zone, including mile posts at whole number values. And by simple addition, such an infinitesimal cloud can be shifted to the neighbourhood of any particular value we please.

    What?

    I’m sorry but your use of terms, your misinterpretations of common, accepted mathematical concepts, constructs and procedures and your inability to answer daveS’s question after repeatedly being asked to do so makes this discussion extremely frustrating.

    I get that you think that either gradually getting bigger or gradually getting smaller eventually ‘catapults’ you into the ‘transfinite’ but that is just not true. Sets can be infinite, values arrived at by stepwise increments are not.

    If you want to deal with hyper-real numbers then you’d best do so properly and using standard arguments and notation.

  468. DS, My point is that there are endlessly more onward members than any specific finite value as proved not merely asserted. Just one more is not enough. And in a context where symbols are loaded, answering y/n to a loaded question is not sensible. The difference in context has to be explained. And that is in part why I am looking to the concrete case to illustrate many features of the problem. If you cannot see why there are endlessly remote rows of the sequence beyond any finite k no matter how large on the primary sense of infinite, then there is a problem. Long before we parse: for all n there exists . . . and get into debates on the existential import of all vs there is at least one etc. Which goes all the way back to the issue that, suitably understood, there is validity yet to the classic square of opposition, cf the discussion at SEP. KF

  469. In kf’s long repeat of things said many times before, and of things we agree about, such as the endless nature of the natural numbers, he says,

    So, ordinary induction shows that we can chain a conclusion in a do forever, but always finite loop. It will be reliable for any finite value, but to point across the infinite endless ellipsis and conclude all natural numbers are finite is tantamount to saying that the endless tapes are endless but have only finitely remote points. But beyond ANY finite value no matter how high, there is an endless continuation.

    That endlessness does not vanish into finitude, it is there, and by force of logic there is a remote zone beyond any arbitrarily large but finite row number. One that we can in effect secondarily model as transfinitely remote.

    This encapsulates the confusion. We all agree that there are are an infinite, endless number of further natural numbers past any arbitrarily large specific natural number k. However, I’m thinking the concrete metaphor of “endless tape” is perhaps confusing things, because there could be no real endless anything. I think we all understand well enough to couch things in strictly mathematical terms.

    We all agree, I think, that

    1. The set of natural numbers is infinite (the tape is “endless”), the primary reason being that it can be put in a 1:1 correspondence with proper subsets of itself.

    2. For any natural number k, there are an infinite number of further natural numbers greater than k. (“beyond ANY finite value no matter how high, there is an endless continuation”)

    3. Any particular natural number k is finite.

    Here’s where we disagree:

    kf says, that to conclude that “all natural numbers are finite is tantamount to saying that the endless tapes are endless but have only finitely remote points”, and he rejects this conclusion.

    However, this is the conclusion we hold. Adding to statements 1, 2, and 3, above, we add

    4. All natural numbers are finite.

    Instead, kf says

    by force of logic there is a remote zone beyond any arbitrarily large but finite row number. One that we can in effect secondarily model as transfinitely remote.

    Now one can interpret this in two ways. If he means that for every k there are an infinite number of further natural numbers greater than k, then he is merely repeating point 2 above.

    However, he seems to mean something more. In Dave’s language (minus the tape metaphor) kf seems to be asserting that

    5. There exists a number X (or numbers) in the set of natural numbers such that for every natural number k, X is greater than k.

    X would then be a number in the “transfinitely remote zone”, and would be an example of a natural number that is not finite.

    Given that kf says his conclusion follows by “force of logic”, it would be appropriate to focus on simple statements such as I have presented in order to follow his logic.

    So, simple question to kf. Does statement 5 adequately represent, in mathematical language, what you mean when you say “there is a remote zone beyond any arbitrarily large but finite row number. One that we can in effect secondarily model as transfinitely remote.”

  470. EZ,

    I am actually busy, but will say this much, it would be wise to read the thread above before jumping in to comment. If you do, you will see the point of surprise when the Wolfram listing of the succession of ordinals appears; I set apart the list and bold it:

    http://mathworld.wolfram.com/OrdinalNumber.html

    In formal set theory, an ordinal number (sometimes simply called an “ordinal” for short) is one of the numbers in Georg Cantor’s extension of the whole numbers. An ordinal number is defined as the order type of a well ordered set (Dauben 1990, p. 199; Moore 1982, p. 52; Suppes 1972, p. 129). Finite ordinal numbers are commonly denoted using arabic numerals, while transfinite ordinals are denoted using lower case Greek letters.

    It is easy to see that every finite totally ordered set is well ordered. Any two totally ordered sets with k elements (for k a nonnegative integer) are order isomorphic, and therefore have the same order type (which is also an ordinal number). The ordinals for finite sets are denoted 0, 1, 2, 3, …, i.e., the integers one less than the corresponding nonnegative integers.

    The first transfinite ordinal, denoted omega, is the order type of the set of nonnegative integers (Dauben 1990, p. 152; Moore 1982, p. viii; Rubin 1967, pp. 86 and 177; Suppes 1972, p. 128). This is the “smallest” of Cantor’s transfinite numbers, defined to be the smallest ordinal number greater than the ordinal number of the whole numbers. Conway and Guy (1996) denote it with the notation omega={0,1,…|}.

    From the definition of ordinal comparison, it follows that the ordinal numbers are a well ordered set. In order of increasing size, the ordinal numbers are

    0, 1, 2, …, omega, omega+1, omega+2, …, omega+omega, omega+omega+1, ….

    The notation of ordinal numbers can be a bit counterintuitive, e.g., even though 1+omega=omega, omega+1>omega. The cardinal number of the set of countable ordinal numbers is denoted aleph_1 (aleph-1).

    I trust this will be enough to give some pause in the rush to dismissive judgement.

    KF

  471. Hi EZ. This sequence has very little to do with the issues about the natural numbers.

    kf accepts that N is an infinite set, and that w and aleph null are names for the transfinite ordinal and cardinal numbers associated with N. He also accepts that w is not a number within N, but rather a number about N: the number of natural numbers within N. By accepting w, he obviously accepts that there are an infinite number of natural numbers.

    That’s about all you need to know from previous posts in respect to the current discussion about the natural numbers.

  472. #470 KF

    And in a context where symbols are loaded, answering y/n to a loaded question is not sensible. The difference in context has to be explained. And that is in part why I am looking to the concrete case to illustrate many features of the problem.

    It’s mathematics, symbols aren’t ‘loaded’. This isn’t an ideological disagreement.

    You’re just being asked a question which you can give a yes or no answer to.

    472 KF

    Yes, I know what ordinal numbers are! You can be very, very patronising at times.

    You don’t get to the first ordinal number by counting up from 0 in steps of one!! Just because the set of ordinals is well ordered doesn’t mean you can get from the finites to the infinites in a step-wise fashion!! There are an infinite number of whole numbers before you get to omega so you can’t get there by counting! Neither can you get there via getting step-wise smaller and smaller.

    Again: omega is larger than any particular whole number but you can’t climb the staircase of whole numbers and suddenly step onto an infinite step. It doesn’t work that way. Every whole number has another whole number 1 bigger than it. Every step up lands you on another finite step and you CAN count how many steps it took to get there.

    This is not context driven or ‘loaded’. Either you get it or you don’t. Not everyone gets it and there’s no shame in that. I don’t ‘get’ Donald Trump or Kim Kardashian, they’re both just so much gobbly-gook to me.

  473. 473 Aleta

    kf accepts that N is an infinite set, and that w and aleph null are names for the transfinite ordinal and cardinal numbers associated with N. He also accepts that w is not a number within N, but rather a number about N: the number of natural numbers within N. By accepting w, he obviously accepts that there are an infinite number of natural numbers.

    Are you sure he accepts that? Because he seems to think he can all of a sudden take another step and get to the infinites. I think he’s mixed up about what well ordered means.

    Anyway, we’re not going to change his mind so it’s probably time to quit.

  474. KF,

    DS, My point is that there are endlessly more onward members than any specific finite value as proved not merely asserted.

    The question was not whether this statement is proved or assumed.

    The question was which of my two statements in #419 you were making.

    Finally it appears that you are clearly stating #1, which comes as a surprise to me. Just a couple of days ago, I said I was 80% sure you were making statement #2.

    And in a context where symbols are loaded, answering y/n to a loaded question is not sensible.

    Well, as you have now clarified that it’s 1 and not 2, it appears that it was/is sensible in this case to ask for yes/no answers.

    If you cannot see why there are endlessly remote rows of the sequence beyond any finite k no matter how large on the primary sense of infinite, then there is a problem.

    Pardon, but the problem was your lack of clarity, not my understanding.

    And despite your slipping back into ambiguity, I will interpret this sentence as again affirming 1 and not 2.

    ***

    Now that we have settled that, let’s observe that statement 1 is consistent with the proposition that all natural numbers are finite. And of course that each cell C in the infinite Turing Machine tape is finitely many steps from cell 0.

  475. Aleta, I simply point to the two tapes, with rows endlessly beyond any row that is finitely beyond the near end. Infinity, primary sense. At least, a break from tax day here. KF

  476. Yes, we agree on this: for any natural number k there is an infinite number of natural numbers greater than k. That is what it means to say the set is infinite.

    “Simply pointing” to this is just a restatement of something we all agree on. I don’t think you need to keep repeating this.

    What is not clear is your beliefs about points 4 and 5 in post 471:

    Are all natural numbers finite? You claim this is not true.

    Or do you believe, as 5 says, that “there exists a number X (or numbers) in the set of natural numbers such that for every natural number k, X is greater than k. In such case, X would then be a number in the “transfinitely remote zone”, and would be an example of a natural number that is not finite.”

    If all natural numbers are not finite, you seem to be claiming that there are natural numbers that are something other than finite. Please explain that in more precise language.

  477. Aleta, further to this, the conditions set up that there are endlessly remote cells in a zone beyond any finitely remote k no matter how large. If we set up a scheme that cannot accept and comfortably digest this primary sense of the infinite, something is deeply in need of rethinking. Especially as we can show the significance of k, k+1 etc by putting the blue tape onward from row k in 1:1 correspondence endlessly with the unmoved pink tape from 0, 1, 2 etc. This helps make my concerns concrete. KF

  478. EZ, if you think that there are not loaded issues in Math then you know little of the relevant history. Was it 20 years ago a suit from C18 was settled between families of two mathematicians? In this field cf Hilbert et al vs Cantor, and the issues Cantor had with others also. In the mix put Russell’s paradox and the import of the setting up of ZFC. Toss in Godel on incompleteness for good measure. What I am saying is there are some concept gaps and that is why I use the thought exercise of punched tape to focus issues concretely. And it is why I put up an algorithm at 217. KF

  479. so, kf, you agree with this statement (which is an attempt to put into more precise mathematical language whta you are saying:)

    “there exists a number X (or numbers) in the set of natural numbers such that for every natural number k, X is greater than k. In such case, X would then be a number in the “transfinitely remote zone”, and would be an example of a natural number that is not finite.”

  480. #480 KF

    You forgot to mention Newton and Liebniz. Aside from the 18th century disagreement I am not familiar with, the rest of your examples have been vigorously argued over and discussed but no one was persecuted or ostracised with the possible exception of Cantor. I would argue that Cantor had other problems which made it harder for him to deal with the academic animosity his ideas created. Make no mistake, what Cantor proposed was incredibly radical and such ideas have to be tested in the fires of scrutiny. As they should. He made an extra-ordinary claim which required extra-ordinary proof and some time to sink in. Now his ideas are not controversial. And I think you’ve erroneously thrown in Godel’s incompleteness theorem which is not the mathematical equivalent of Heisenberg’s uncertainty principle.

    Of course there are some gaps and controversies but these days a) they are worked out in a collegiate fashion and b) what we are discussing with you is not controversial any more. Just listing some of the past ‘controversies’ does not prove your point that issues are ‘loaded’. It just means that people disagree usually for very good reasons.

    In mathematics the ‘battleground’ is now purely intellectual and academic. The only ‘loading’ is egotistical. No one is marginalising others, no one is being forced out of academic positions, no one is being stifled or bullied.

    My point is that using the term ‘loaded’ is inaccurate.

    Our disagreement with you is purely mathematical. There’s no call to add a layer of manipulation or menace.

  481. Aleta, the issue is the tape, always the tape. If we cannot satisfactorily model something as simple as the thought exercise of an endless punch paper tape, we do not genuinely understand yet. This includes that we must be able to speak to the zone that is far away receding endlessly from us, with punched holes in it. Where beyond any arbitrary finite k, however large (I picked 10^150 and 300, the square of that, to give and idea of what I am saying), there will be endlessly more rows. Such that we may pull in k, and do so k times and yet still there is going to be endless tape remaining yet ahead such that the remainder may be matched 1:1 with the un-pulled tape next to it. And pardon me but the term I used is not meant to suggest hidden agendas and ideological threats or stratagems, only that terms are freighted — is that less suggestive? — with perceptions and contexts. The issue is I think there is a paradigms issue here. KF

    PS: Note my concern is pivoting around the claim there is an infinite number of finitely large natural numbers, where any finite k — ponder the tape to see why — will not be transfinitely remote, endlessly remote; k* 0.1 inches will be exceeded and bounded by (k+1)*0.1 inches and so forth on a do forever. Any finite k can be arrived at in k finite steps of +1 from 0, then exceeded in the next step, with +1 here going at the rate of 0.1 inches per step. This is the context in which I looked again at what say ordinary mathematical induction actually shows, and what is entailed by the apparent pattern of do forever loops starting in axioms, and the commonplace ellipsis of endlessness.

  482. Again, you repeat something we agree with:

    Where beyond any arbitrary finite k, however large (I picked 10^150 and 300, the square of that, to give and idea of what I am saying), there will be endlessly more rows.

    That is, for any arbitrary finite k, there are an infinite number of further natural numbers greater than k.

    WE AGREE WITH THIS!!! Do you get that!!! WHY DO YOU KEEP REPEATING THIS???

    If all you mean by “remote zone” is this fact, that there will always be an infinite number of further numbers, then you are not saying anything that we don’t all know.

    But what you are NOT doing is responding to a simple argument:

    1. The set of natural numbers N is an infinite set.

    2. Every specific natural number k is finite.

    3. So either, by force of logic, every one of the infinite number of natural numbers in N is finite (our position), or there are numbers in N which are not finite (which seems to be your position.)

    Which is it, then: is every natural number of finite, or are there numbers in N that are not finite.

  483. #483 KF

    Note my concern is pivoting around the claim there is an infinite number of finitely large natural numbers, where any finite k — ponder the tape to see why — will not be transfinitely remote, endlessly remote; k* 0.1 inches will be exceeded and bounded by (k+1)*0.1 inches and so forth on a do forever. Any finite k can be arrived at in k finite steps of +1 from 0, then exceeded in the next step, with +1 here going at the rate of 0.1 inches per step. This is the context in which I looked again at what say ordinary mathematical induction actually shows, and what is entailed by the apparent pattern of do forever loops starting in axioms, and the commonplace ellipsis of endlessness.

    I give up. You’ve got some issue that I can’t discern which is throwing up a roadblock.

    Please feel free to stand in opposition to well established mathematics. Because you are not a research mathematician it probably doesn’t matter anyway. If you’re teaching students then I do worry but I can’t do anything about it.

  484. Aleta,

    3. So either, by force of logic, every one of the infinite number of natural numbers in N is finite (our position), or there are numbers in N which are not finite (which seems to be your position.)

    Which is it, then: is every natural number of finite, or are there numbers in N that are not finite.

    By the rules of right reason, as a matter of fact! 🙂

    Surely KF cannot avoid answering this question?

  485. EZ,

    again is or is not each tape receding to the endlessly remote. If not the tape is finite. If it is, it is infinite and there will be rows infinitely remote. Infinite in the primary sense.

    These cannot be reached by any cumulative process of finite stage steps but on attaining the potentially infinite we may point to the endlessly remote zone. As is routinely done in practical mathematics.

    A look at the ordinary mathematical induction shows that it embraces a do forever loop in steps from case k to k+1 in succession. Also, when we look at the set of natural counting sets in succession we see:

    {} –> 0
    {0} –> 1
    {0,1} –> 2

    . . .

    or,

    {0,1, 2 . . . k, k+1 . . . Ellipsis of endlessness . . . }

    That is there is a span of endlessness within the set, which by definition cannot be bridged in finite steps.

    That’s where I would expect to find a far zone that is ideally there but cannot be finitely attained to in steps.

    Where for every finite k, regardless of how large, k is bounded by k+1 and is finite and attainable in k +1 increments from 0.

    The ordinary induction on case 0 or case 1 and the chaining principle Case-k => case-k+1, is inherently finite though open ended.

    Whatever model we need to account for our tape as a thought exercise needs to account for these phenomena.

    And sorry, this is mathematics, appeals to modesty in the face of collective authority are not enough.

    There is a thought exercise on the table, close to the traditional Turing Machine.

    We need a reasonable scheme that adequately accounts for an endless tape that recedes from row 0 to the RHS with rows of holes all the way. Actually, two, one pink, one blue. The blue is pulled in k rows, k times over. Due to endlessness, at each k-pull, it must still match pink 1:1 because of endlessness.

    Which entails that there are infinitely many rows, accumulating at 0.1 inch per row to the RHS:

    |0, 1, 2 === . . . k, k+1 [finite values] . . . ===> . . .

    |k, k+1 [finite values] . . . ===> . . . (matching 1:1)

    KF

  486. DS, I am trying to see how it can be reasonably concluded that there are infinitely many finite natural numbers. The two tapes thought exercise is in that specific context. To my sense, it would seem hard to avoid that every finite k is reachable in k finite steps and is exceeded on the k+1 step, so no span from 0 of finite stages in cumulative succession can be transfinite, can end the endless. Therefore I incline that the sets of ordered counting sets in endless succession should reach a zone which each member will itself be endless. What such is labelled or classified as is secondary. I could see with the naturals being defined on an unending succession of finite steps where every number actually attainable in finite-stage increments to k number of steps will be finite [where k may be arbitrarily large but not transfinite], but that points beyond to the endlessness. KF

  487. KF,

    DS, I am trying to see how it can be reasonably concluded that there are infinitely many finite natural numbers. The two tapes thought exercise is in that specific context. To my sense, it would seem hard to avoid that every finite k is reachable in k finite steps and is exceeded on the k+1 step, so no span from 0 of finite stages in cumulative succession can be transfinite, can end the endless.

    The Turing Machine tape is an excellent thought experiment. And yes, it is true that every finite k is reachable in k finite steps.

    It is also true that no “span” from 0 consisting of finitely many steps can be transfinite. In other words, {0, 1, 2, …, k} is always a finite set.

    Therefore I incline that the sets of ordered counting sets in endless succession should reach a zone which each member will itself be endless.

    I don’t know why this would be necessary.

    The collection of finite “counting sets” is “endless”, no?

    I could see with the naturals being defined on an unending succession of finite steps where every number actually attainable in finite-stage increments to k number of steps will be finite [where k may be arbitrarily large but not transfinite], but that points beyond to the endlessness. KF

    Yes, the first part describes the natural numbers accurately. In fact, the Peano axioms specify that each natural number is obtainable by applying the successor operation to 0 finitely many times, so you cannot generate any members of N at infinite distance from 0.

  488. EZ & Aleta:

    (& DS, attn HRUN . . . who needs to learn what has been pointed out several times but studiously ignored: that Mathematics sometimes advances by challenging the consensus so appeal to authority rather than the logic of structure and quantity, fails . . .),

    kf, I actually understand perfectly well and you just reaffirmed it. You are right and DS is wrong. And if math disagrees with you it is obviously wrong as well. There is no appeal to authority, just a description of the situation from your point of view.

  489. I give up also. See my post at 462 where I summarize the ways in which kf avoids specificity, repeats things we agree on, and will not answer direct questions that would help us understand what he means (such as a very simple question at 484).

    But I’ve enjoyed conversing with dave and ellazimm! 🙂 Thanks.

  490. Folks, I again simply point to the tapes thought exercise. If every value for the number of a row in the succession in the endless tapes is finite, the value will necessarily be some k (i.e. kth row), exceeded by k+1 and achieved in k increments of 0.1 inches. This finite value cannot be endless, being completed in k steps and then bounded and exceeded by k+1. How then is the span of the tape with rows every 0.1 inches along its length in succession from row 0, endless, apart from that for any finite k, there will be k+1, etc onward without limit, and by limitlessness violating the claim that every value corresponding to a natural number in the sequence 0, 1, 2 . . . without end is finite? And no I am not pretending to be cleverer than all Mathematicians etc, I am asking how is an apparent paradox to be resolved without falling into contradiction? KF

  491. re 490: I am appealing to logic, and kf is failing. See 484.

  492. KF,

    I’m getting lost in that italicized sentence.

    It might be helpful to list the assumptions, one by one, which supposedly lead to a contradiction.

    For example:

    1) The tape is endless (that is, infinite).

    2) The rows are each labeled with finite natural numbers in increasing order (so row 0, row 1, row 2, etc).

    3) The “span” of the tape is endless (?) I don’t really know what this means, other than there are rows in the tape arbitrarily far apart. For example, there are rows 10^150 inches apart. There are also rows 10^150^150 inches apart. There are rows any finite number of inches apart.

    And so on.

  493. kf has several times written whole posts about the importance of basic logic as a foundation for right reason. And yet, when faced with a simple logical statement (either all natural numbers are finite or some natural numbers are not finite), he refuses to address it.

    This has nothing to do with red and blue tapes, or with the fact that we all agree that there is an endless number of natural numbers past any particular finite number k, or about authority.

    This is just, as kf says, about logic and structure. Address the logic, kf!

    Is it true that one or the other of these must be true: either all natural numbers are finite or some natural numbers are not finite?

    Stay true to your principles here and address the logic.

  494. Aleta,

    There is a logical issue at stake, and when you and DS asked me about it, I replied in logical terms.

    Cf 487 – 8 & 492.

    KF

    PS: Let me borrow from 497 below:

    If the tape goes on endlessly beyond any arbitrarily large value (and that is required to retain 1:1 matching) then as the rows have a finite pitch of 0.1 inches, rows are not merely arbitrarily far apart. For, it seems that between the zone we can attain to in finite +1 steps and the far zone of endlessness there will be rows that are transfinitely far apart, endlessly far apart.

    This does not seem to be consistent with the argument that as case 0 is finite and case-k being finite entails case-k+1 is finite, then ALL cases for rows will be finite in label and finitely remote.

    At minimum, there is a paradox there to be resolved. Or at least, that is how it seems.

    Next, it seems to me that inherently the sort of ordinary induction just outlined itself crucially relies on +1 step chaining and is subject at every step to the onward unreachable endlessness. Moving to the assumption that cases b less than a case a are all so entails case a for all a in A [as a new form of accumulation], with less than only implying a built in ordering relationship, then leads to the issue as to the span of A.

    (Is A finite or transfinite and if the latter does it necessarily enfold only finite members, or is there an implicit involvement of the transfinite in individual members?

    In effect, where do we close off the curly braces, and what does this entail when an ellipsis of endlessness is involved or implied?

  495. DS,

    The basic point is endlessness. In effect, pick any arbitrarily large value k, pull in the blue tape by k rows, k times. Every one of those k times the remaining tape will still match the unpulled tape 1:1, providing both are endless. That is the sets of rows on both tapes are infinite.

    The paradox I perceive comes in in claiming at the same time that EVERY row corresponds to a finite ordinal value.

    Let that value be k, it can be attained from 0 in k steps, and exceeded in the k+1th. That can be seen for any arbitrarily large finite row number, but the problem is the tape goes on endlessly beyond that, by definition and with rows punched in all the way.

    Something we cannot attain to in steps.

    Represented so innocently by three dots in an ellipsis.

    (And the one who, above, dismissed the use of y = 1/x as what I have called a catapult needs to look to the discussion of hyper reals and infinitesimals in nonstandard analysis. I just pick up another point, it seems there is little point in replying to every dismissal when given in such sharp terms.)

    If the tape goes on endlessly beyond any arbitrarily large value (and that is required to retain 1:1 matching) then as the rows have a finite pitch of 0.1 inches, rows are not merely arbitrarily far apart. For, it seems that between the zone we can attain to in finite +1 steps and the far zone of endlessness there will be rows that are transfinitely far apart, endlessly far apart.

    This does not seem to be consistent with the argument that as case 0 is finite and case-k being finite entails case-k+1 is finite, then ALL cases for rows will be finite in label and finitely remote.

    At minimum, there is a paradox there to be resolved. Or at least, that is how it seems.

    Next, it seems to me that inherently the sort of ordinary induction just outlined itself crucially relies on +1 step chaining and is subject at every step to the onward unreachable endlessness. Moving to the assumption that cases b less than a case a are all so entails case a for all a in A [as a new form of accumulation], with less than only implying a built in ordering relationship, then leads to the issue as to the span of A.

    (Is A finite or transfinite and if the latter does it necessarily enfold only finite members, or is there an implicit involvement of the transfinite in individual members?

    In effect, where do we close off the curly braces, and what does this entail when an ellipsis of endlessness is involved or implied?

    In the case,

    {0,1,2 . . . k, k+1, . . . [ellipsis of endlessness . . .]}

    that seems to entail order type w, cardinality aleph null and that assigning w [omega] to the whole

    {0,1,2 . . . k, k+1, . . . } –> w

    does not suddenly introduce a new property on the RHS. It is there on the LHS already. That is, the span is endless and implicitly, members in the far zone from 0 will hold endlessly large values. As, each member from 0 on in succession is the collection of the preceding members so endlessness of the whole will be attained by endless collection in particular members. The incrementally emergent set in effect self copies its members so far endlessly to create new members, until we point onward from the potentially infinite to the whole and somehow symbolise the whole. At least, that is how it looks. And that means that w does not suddenly emerge as the follow on to any particular member but that it is emergent on pointing to endlessness as completing step. By use of the ellipsis. It is a conceptual leap on recognising endlessness. At least, again, that is how it looks.)

    That is how it looks, not a happy picture [e.g. a set incrementally swallowing itself like the proverbial snake to emerge anew as extended to successor is instantly uncomfortable as we look to the set as a whole endlessly continued], but that is the picture I see.

    KF

  496. #491 Aleta

    But I’ve enjoyed conversing with dave and ellazimm! 🙂 Thanks.

    You are a star. With a lot more patience than me.

    I find the psychology of debates on this sight very interesting. You and I are used to being in an academic situation where you get things wrong at times. And we’ve learned to own up to our mistakes and move on. But if you’re not used to that (and if you feel like you’re besieged on all sides) then you can’t afford to concede on anything. You’re constantly afraid that any hole in the damn might become a flood that will drown you. I get that but, in this case, we are just talking about objective mathematics. I have no intention of wading into the question of the existence or non-existence of an infinite past for the universe. And the solution to that conundrum is a matter of using the correct model NOT the mathematics underlying the model. So KF’s basic approach (attack the mathematics to uphold the world view) fails at the gate. And he’s wrong about the mathematics as well.

    But he’s in ‘circle the wagons’ mode. And his real ‘battle’ has nothing to do with mathematics.

  497. All good points, EZ, and I too find the psychology interesting. I especially agree with you about the difference between pure mathematics and the task of modeling the world with mathematics – I’ve had some long conversations here about that before.

    For what it’s worth, I’m a bit interested in my own psychology – why do I keep posting??? I’m sure I’m close to an end.

  498. to kf. I’ll note that you repeat, again, all sorts of things you’ve said many times before.

    But you won’t answer a simple and obvious question.

    Is it true that one or the other of these must be true: either all natural numbers are finite or some natural numbers are not finite?

  499. KF,

    Let that value be k, it can be attained from 0 in k steps, and exceeded in the k+1th. That can be seen for any arbitrarily large finite row number, but the problem is the tape goes on endlessly beyond that, by definition and with rows punched in all the way.

    Something we cannot attain to in steps.

    Perhaps not, but who said it could? [Edit: I’m assuming beginningless processes are off the table here!]

    There are many mathematical objects that cannot be constructed in a finite (or even countable!) number of steps. The real numbers are even worse.

    Most mathematicians don’t envision even the natural numbers as being constructed in a sequence of steps. There’s the Axiom of Infinity, and that’s that.

    If the tape goes on endlessly beyond any arbitrarily large value (and that is required to retain 1:1 matching) then as the rows have a finite pitch of 0.1 inches, rows are not merely arbitrarily far apart. For, it seems that between the zone we can attain to in finite +1 steps and the far zone of endlessness there will be rows that are transfinitely far apart, endlessly far apart.

    Specifically which rows are infinitely far apart? You have natural number labels for each row, so if this were the case, wouldn’t you be able to name some of these rows? In the sentence: “Row 0 and row ____ are infinitely far apart”, what number goes in the blank?

    In reality, there is just one “zone”, consisting of all rows finitely distant from row 0.

    I don’t follow most of the rest, but this:

    And that means that w does not suddenly emerge as the follow on to any particular member but that it is emergent on pointing to endlessness as completing step. By use of the ellipsis. It is a conceptual leap on recognising endlessness.

    seems at least partially correct. As we have seen, ω is not a successor to any ordinal. ω is not of the form α ∪ {α} for any other ordinal α.

    It is literally the union of all finite ordinals. Or, put another way, ω = N.

  500. This part of my post #501 needs correction, and conveys the opposite of what I meant:

    There are many mathematical objects that cannot be constructed in a finite (or even countable!) number of steps. The real numbers are even worse.

    For example, I mean in the Cauchy sequence version of the construction of R, you do not iterate through all (classes of) Cauchy sequences in Q one after another and build up R one element at a time, even though that might appear to be the case if you try to conceive if it as a looping process.

    The real proofs and constructions of such objects consist of a finite number of steps.

  501. #499 Aleta

    For what it’s worth, I’m a bit interested in my own psychology – why do I keep posting??? I’m sure I’m close to an end.

    It is like a bag of Doritoes isn’t it? You crave them and then when you eat too many you feel a bit ooky.

    I find the perpetual denialism, especially about things that are not in any way controversial, interesting and perplexing. It’s a real insight into agenda driven thinking and cognitive biases. The whole culture is fascinating.

  502. Yep – more good points. I agree with all you say.

  503. A parting shot: kf will not answer my direct questions, we know, so I am going to answer them for him.

    The issue is this: we claim all natural numbers are finite. kf does not agree this is so.

    We all agree that the set of natural numbers N is infinite. We agree that means that for any natural number k there are an endless, infinite number of further natural numbers. In fact it is this property – the fact that we can put any proper subset of the natural numbers in a 1:1 correspondence with the set of natural numbers itself, that is the characteristic that defines the infinite nature of the natural numbers.

    We all agree about all of the above paragraph.

    Now there are only two logical possibilities:

    A: Every natural number is finite, or

    B. There are natural numbers that are not finite.

    kf rejects A.

    Therefore, kf concludes B: there exists non-finite natural numbers.

    That is, he believes there is a least one, or more, numbers X which are greater than any finite natural number.

    This is consistent with his vague, unspecified language about “passing the ellipsis” into the “far zone”, which is “infinitely far from” all finite natural numbers.

    This is what kf believes: he believes in the existence of non-finite natural numbers that are greater than any and all finite natural numbers.

    He believes in a kind of transfinite number within the set of natural numbers. For kf, N = {1, 2, 3, … k, k + 1 (the finite naturals) … the transfinite naturals}

    This is, I think, the inescapable conclusion to be reached from discussing this issue with him for quite a lengthy time.

  504. Aleta,

    I think you’re right. I had a glimmer of hope when I read this from KF:

    DS, My point is that there are endlessly more onward members than any specific finite value as proved not merely asserted.

    but things seem to have reverted.

    After spending quite a bit of time in these threads, I found it interesting to look at this post, which begins:

    Crimnologist and former atheist Mike Adams summarizes the three foundational philosophical alternatives to the Cosmos:

    First, we can say that it came into being spontaneously – in other words, that it came to be without a cause. Second, we can say that it has always been. Third, we can posit some cause outside the physical universe to explain its existence.

    We are here struggling to understand a paper tape (albeit infinite), and they are over there solving the origin of the cosmos.

    Well, I don’t think you or ellazimm or I are having any difficulty with the tape, and I find their freeswinging origins arguments unpersuasive at best, but it’s an interesting contrast.

  505. #506 daveS

    Well, I don’t think you or ellazimm or I are having any difficulty with the tape, and I find their freeswinging origins arguments unpersuasive at best, but it’s an interesting contrast.

    The only trouble I had with the tapes stuff was why it was introduced. I guess it was some example KF found and, in his usual copy-and-paste style he threw it in to make the discussion sound more . . . academic. I tend to stick with KISS myself when trying to explain complex things. Maybe he’s used to being able to convince people by throwing wave after wave of hard to understand stuff and eventually they agree with him even if they really don’t understand what he’s talking about.

    Anyway, we seemed to have scared KF off. He hasn’t been on UD at all since his last comment above. Maybe he’s just busy.

  506. Aleta,

    IIRC, I am the one guilty of bringing in the Turing machine tapes, just because I thought they illustrated the Hilbert Hotel (and the set N) well. I also thought they would enable KF to understand our point of view, but unfortunately that didn’t happen. I brought up the tapes sometime after KF made reference to the PA system (!) in the Hilbert Hotel, and I was looking for a way to strip away all the irrelevant implementation details.

  507. No guilt, of course!

    But I do think the concrete nature of the tapes has helped fuel kf’s confusion. As we discussed previously, the tape reinforces the idea that the natural numbers must be traversed one-by-one to come into existence, rather than already existing in their totality by virtue of the definitions.

    Concretely, there can be no endless tape in any material sense: the tape is endless in the sense that the process of unrolling it further can never stop, but thinking of a concrete thing being infinitely long conflates the metaphor with the abstraction.

    So I think the tape has been a confusing metaphor.

    Also, and I’ll mention this now that the discussion is over, I also think a source of kf’s confusion is that the beginning stimulus of the topic concerned time as a model for the natural numbers (the integers, actually.) Two problems with this: one is that we envision ourselves, or the world, as being in time, moving through it step by step, and that we see ourselves as only being able to move in one direction. So time, like the tape, emphasizes the view from any particular point rather than a mathematical view of the set as a whole.

    Ironically, what I am saying is that, rather than a view from inside the natural numbers, mathematically we have a “God’s eye” view of the set as a whole, from outside the natural numbers. This is what Cantor did when he declared/created w and aleph null as transfinite numbers: he looked at the whole infinite set of natural numbers and began doing mathematics with its infinitude as a concept in and of itself.

    And, to perhaps further the irony, and the idea of a God’s eye view, how would God see the natural numbers? Given that God is supposed to be outside of time, and infinite in scope, would he have any problem seeing that all the natural numbers were finite, and yet at the same time be able to see the entire infinite set of them? 🙂 Supposedly God can both see the world from any particular moment in time, now, and simultaneously see all moments of time: all the past and all the future. If he can handle that with the entire universe, surely he cold handle it with the natural numbers.

    P.S. This last point is of course not a real topic of conversation, but I think it is interesting to perhaps tie the topic of mathematical infinity to the speculative theological beliefs of kf and others here at UD. KF just wrote in the thread that Dave mentioned that “If a cosmos now is, something always was, a root of the reality we experience.” So theologically kf is willing and seemingly able to accept the existence of an infinitude as a whole, from a God’s eye point of view, in respect to “the root of reaity”, but not so able mathematically when contemplating the infinitude of the set of finite natural numbers.

    Just some thoughts.

  508. A theological question: Does God need an ellipsis?

  509. Aleta,

    Good points. I hadn’t thought how the tape model could reinforce these misconceptions, but I think I understand now.

  510. #511 daveS

    Good points. I hadn’t thought how the tape model could reinforce these misconceptions, but I think I understand now.

    I wouldn’t worry about it, KF has an agenda and for various reasons he was bound to disagree with something somehow.

    If you check his own website stuff you’ll see how he can mangle even straightforward mathematics. His motivations aren’t about the mathematics but about what he can support with what he thinks he understands.

    #510 Aleta

    A theological question: Does God need an ellipsis?

    Well, if s/he/it is the alpha and omega, the beginning and the end then sometimes you must find a way to fill in the intermediate space!

    More seriously, it’s we who need the ellpsii. Ellipsises? Whatever. But really they’re just short hand for “and so on in the same fashion”. Like much of mathematics, we replace thoughts and procedures with symbols ’cause they’re easier to write down.

    Get your students to write out entirely in words a simple algebra problem sometime.

  511. Aleta, EZ & DS:

    I notice, a pattern that needs to address an underlying structure in Mathematical thought. For, there is a major shift of approach that happens when one moves to axiomatic systems. Such in effect set up abstract model worlds that may be fruitful but — ever since the 1930,s — face Godel’s two point challenge. For no suitably complex system will we have both consistency and completeness, and there is not a constructive approach that guarantees consistency even at the price of limitations.

    Such is fundamental, and one of the implications is that test cases (in Mathematical contexts, often abstract thought exercises) are important. In the general area being disputed in the above thread, Russell’s paradox on sets (and the illustration of the village Barber for whom as he shaved himself it was undecidable to assign him to the set choices shave self vs shaved by Barber leading to fatal ambiguity) forced reformulation. Similarly, Hilbert’s Grand Hotel Infinity was a challenge to understand implications of the concept, infinity.

    A, I must note to you that you are resorting to refusing a given answer stated in the explicit context that there is a gap in views that can make simplistic y/n without explanation meaningless.

    I have put up the case of the pink/blue punch tapes running off endlessly precisely to put on the tape a primary, concrete case. In that on pulling in the blue tape any arbitrarily large but finite number of rows, k, the remaining tape from k, k+1 etc on can be put in 1:1 correspondence with the unshifted pink tape. Indeed, the exercise may be done k times over, each time j showing that the j-1th try has left the endlessness intact and so empowering an ordinary mathematical induction that any finite pull in does not terminate endlessness and so also the property of being infinite. By generally accepted principle, infinite sets of the relevant class can be put in 1:1 correspondence with proper subsets.

    Where, by definition of the tape, all along their length they have punched rows. Accordingly, it is hard to reject the point that there are spatially endlessly remote rows which duly have row-counts that are endlessly larger than any arbitrarily high but finite row value k. Which is the long way round to, these are infinitely far away. With appropriate row numbers.

    How does that fit with endless succession?

    That becomes an issue for paradox, as we see that ordinary mathematical induction starts with case-0 or 1 then adds a chaining pattern of implication and typically projects to all cases, pointing across an ellipsis of endlessness. My concern here is this point across is carrying all the weight of the conclusion, and a more conservative statement would be that all we can reach by finite stage, step by step processes will carry the stepwise propagated property.

    The point of the ellipsis of endlessness is in part that there is a far zone that cannot be spanned in finite stage successive steps. In particular, steps of +1.

    The two tapes illustrate in a thought exercise, how that result comes across.

    In the context of the various axioms, it has been pointed out above, that the set of axioms embeds stepwise processes that get to a potential infinity bridged by pointing across an ellipsis of endlessness. For convenience, Wiki discusses the Axiom of infinity:

    In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x [–> in context the succession from {} on thus far is in mind in a generalised case] with its singleton {x} [ –> this goes on to the next step] is also a member of I. Such a set is sometimes called an inductive set.

    This axiom is closely related to the von Neumann construction of the naturals in set theory, in which the successor of x is defined as x ? {x}. If x is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set. Successors are used to define the usual set-theoretic encoding of the natural numbers. In this encoding, zero is the empty set:

    0 = {}.

    The number 1 is the successor of 0:

    1 = 0 U {0} = {} U {0} = {0}.

    Likewise, 2 is the successor of 1:

    2 = 1 U {1} = {0} U {1} = {0,1},

    and so on. A consequence of this definition is that every natural number is equal to the set of all preceding natural numbers.

    This construction forms the natural numbers. However, the other axioms are insufficient to prove the existence of the set of all natural numbers. Therefore its existence is taken as an axiom—the axiom of infinity. This axiom asserts that there is a set I that contains 0 and is closed under the operation of taking the successor; that is, for each element of I, the successor of that element is also in I.

    Thus the essence of the axiom is:

    There is a set, I, that includes all the natural numbers.

    The axiom of infinity is also one of the von Neumann–Bernays–Gödel axioms.

    In short, there it lies, the potential infinity established by pointing across an ellipsis of endlessness. With von Neumann’s construction closely embedded.

    The concern I have had, then is about that pointing across and its implication in cases like the two tapes.

    One import I see is that the effective definition of the successive counting sets deemed natural numbers makes that set to be the reachable by stepwise do forever successor process that cannot actually be completed due to endlessness. This sets up a paradox between finite stepwise process and endlessness.

    So now we come to labelling and giving values to rows in the tapes. For finite extension in steps that is not an issue. But the onward pointing to endlessness does raise an issue that is not resolved by saying ordinary induction takes them all in in one step. For, as the finite stepwise chain hits any k an endless succession always lies beyond. Bridging by an implicit sub axiom of pointing across an ellipsis of endlessness will be freighting that implicit step with the heavy lifting to reach the conclusion.

    Which leads to the point, why take that step?

    Or, is there not a difference between the primary sense of endlessness of succession and definitions and axiomatic steps that in effect lead to an infinite number/succession of finite, +1 stage separated values, leading to something like infinite finitude. Paradox, it appears.

    Rhetoric about denialism — which is horrifically loaded with Holocaust denial — does not help settle the matter. Nor does the notion, we do not dispute this, it is settled so there, only the ignorant, stupid, insane and/or wicked denialists would challenge the Consensus.

    EZ, I am looking at you here.

    And, A, right from the beginning, I have pointed to the primary sense of the infinite, the endless succession that lurks in the arrow we put on graph axes, origin, line of reals with wholes as milestones marked, arrow of endlessness. Whatever we find by way of axiomatic algebraic model or labels, that succession to endlessness beckons.

    This issue is not some vague mystery that I have been evasive about, I have asked, how can one find a reasonable answer as to how one rejects that an endless succession goes to a zone of endlessness, i.e. beyond finite bounds? And what does this primary, primitive sense have to say to our axiomatisation?

    To date, I find no satisfactory resolution.

    Given say the thought exercise of tapes, I cannot dismiss endlessness, and given the inherent finitude of deeply embedded stepwise +1 stage procedures of succession, joined to pointing across an ellipsis of endlessness, I cannot see that a model that converts the whole number succession into an infinite chain of the finites is less than paradoxical verging on outright contradiction. The finite is ended, the endless by definition is not.

    The endless tape exercise points to a far zone with onward succession not subject to spanning in +1 steps.

    How to capture that in reasonable terms seems unanswered.

    And EZ, motive mongering or how I came up with the idea of pink and blue paper punch tapes is of little value. In fact, as Turing tapes of endlessness were discussed and as coloured 3+5 punched paper computer tapes exist, it seemed reasonable to look at parallel tapes running off endlessly and to ponder in ways that are anchored to an intuitive, simple entity. Thought exercises and abstracted models thereof are commonplace in several linked fields. And there is Hilbert’s hotel.

    Think of the tapes as the inspection report on the HH rooms if you will, a one- ascii character symbol denoting room condition.

    Matters not how they are marked, the point is here are the pink and blue tapes running endlessly to RHS from a zeroth row to begin with. With every row punched in some way.

    Patently, if stretched out with rows going off at 0.1 inch pitch, then if endless there is a remote endless zone just as there is a near ended, zero row neighbourhood zone.

    Concrete and readily represented, even turning glyphs into simple keyboard graphics:

    Pink, with near end:

    |0 === . . . k, k+1 ====> . . .

    Blue, pulled in k and cut just before k, continuing on to RHS endlessly:

    |k, k+1 ====> . . .

    Both can be set in 1:1 correspondence, and pull in k and match can be endlessly repeated beyond case o as above. (I just used do k times to get to k*k rows pulled in.)

    What mathematical language can we use to address that endlessness, resolving paradoxes?

    Obviously, one point is, the endless by the very force of that concept cannot be ended in actuality.

    (And that is not a dubious, dismissible worldview point, it is basic logic.)

    DS, BTW, a PA system vs chaining in sucession on Hilbert’s Hotel brings out the difference between all at once setwide processes and inherently finite chained successive ones. I don’t know about you but instantly on hearing of the hotel and what happens to provide room for fresh guests the issue is, how do you get the guests to all go to room 2n from room n? A PA system, with broadcast capability. By contrast with trying to propagate in a chain of steps.

    Infinite sets dealt with in steps run into the potential infinite then point across the ellipsis of endlessness issue. A broadcast system by contrast operates on the whole at once. And — as fair comment given deteriorated tone at this point as you had exchanges just above — it is a significant conceptual gap that you responded to that with an exclamation mark of dismissal.

    A, we can, broadcast, posit an endless tape with punched rows all along its length, as was done. Then that sets up the issue of stepwise processes vs the transfinite. That allows exploration; which lets us begin to identify the limitations of stepwise processes and to observe how such are embedded from the roots even in axioms.

    Is there anything inherently dubious in positing endlessness all at once, then thinking on how that relates to counting set based approaches? I think not.

    Where, of course, seeing the tape whole and not created by a do forever loop of succession gives a different perspective. Indeed moving to pointing across the ellipsis of endlessness is exactly a pulling back to the original single glance at the whole approach.

    So, where does this all come out?

    For me, it first makes me a lot more wary of discussions of the infinite and the mathematics connected therewith. I have a lot of sympathy for the non standard analysis approach. Hyper reals and infinitesimals. I am now much more wary of discussions of naturals and of reals. When trying to down count from the transfinite runs into headaches so easily, that is a warning flag.

    Tapes, think of a sprocket drive and read head in the far zone decrementing by 1s, -1 steps. Can it reach to a k-neighbourhood of 0 in steps? No, it would have to span endlessness. It can move from some start point and move a finite distance leftwards in steps but absent a definite start point it cannot arrive from the far right zone of endlessness, just as a similar drive and head at k cannot span in steps to the far zone.

    Where of course, easy to discuss concrete thought mechanisms force out things that are easily lost in forests of abstract symbols. That is why thought exercises can be very strategic.

    If we cannot readily resolve one, then that is a sign that we have not got the whole act together yet.

    At the same time, I have a much deeper appreciation for the force of Spitzer’s point. Implicitly ending the endless on successive finite stages is an absurdity and fallacy.

    And, pointing to the tapes, there is need for an adequate discussion of endlessness as a whole not just do forever loops of the potentially infinite completed by pointing.

    KF

    PS: The LOGIC of necessary being and of ontological roots of a contingent world is a point in philosophy not theology. Indeed, it is antecedent conceptually. And the eternal is different from the temporal. The pink and blue tapes thought exercise rests on neither but on glorified common sense abstraction from very real computer technology of several decades past.

  512. F/N: I think I can summarise what seems to make sense to me, for the sake of record.

    First, the two tapes are pivotal, as is the difference between all at once, stepwise +/-1 processes and the sort of convergent series completed in finite spans of space and time that crop up in Zeno’s paradoxes. (For these last, the rapid trend to infinitesimals in time and space converges to a very finite limit in both space and time as a trajectory plays out.)

    Second, the ellipsis of endlessness is crucial to distinguish the near 0 zone and the transfinitely far one. As it cannot be bridged in stepwise +/-1 processes, there is an operational barrier between the zones.

    In that context, it is a plausible step to assign the transfinitely remote zone values such as w, w+1, . . . w+g, . . . so succession continues in stepwise +/-1 processes, but one conceptually catapults to get there. The y = 1/x function applied to mild infinitesimals as a suggestion, would be such a catapult.

    What I can now do is to take

    [a] the inherent finitude of +1 increments from 0, and

    [b] the issue of a do forever loop from 0 that

    [c] goes on endless-LY but never spans the ellipsis of endlessness and

    [d] use the three to synthesise a picture that I think (thus far; this is exploratory . . . ) is coherent.

    The counting succession from 0 in +1 steps is endless in succession process via do forever, but cannot exhaust endlessness. Indeed at any given k, k+1 etc the shifted blue tape can be put back in 1:1 correspondence with the unshifted pink one. This guarantees the span cannot be completed and shows how endlessness beyond any specified finite kth step of arbitrarily large scale, is no nearer to ending the endless than when it all began at 0.

    The succession of such counting steps goes on endlessly but by virtue of that is never completed in the sense of spanning the endlessness. I can see the sense in which the natural counting numbers are defines by that summary and constraint. What we can ever reach is finite but the succession continues, and at any k we then set a bound on k by the k+1th step following. The picture of the natural numbers (and the near 0 zone of the tapes), then would be:

    An endless-LY continued sequence of finite numbers comprising a set that in aggregate — as ideally pointed to across an ellipsis of endlessness — is indeed just such; endless.

    The successor to and order type of that block is then w, followed by its own successors w+1 etc. The far zone of the tapes.

    Thus, per Wolfram:

    http://mathworld.wolfram.com/OrdinalNumber.html

    In formal set theory, an ordinal number (sometimes simply called an “ordinal” for short) is one of the numbers in Georg Cantor’s extension of the whole numbers. An ordinal number is defined as the order type of a well ordered set (Dauben 1990, p. 199; Moore 1982, p. 52; Suppes 1972, p. 129). Finite ordinal numbers are commonly denoted using arabic numerals, while transfinite ordinals are denoted using lower case Greek letters.

    It is easy to see that every finite totally ordered set is well ordered. Any two totally ordered sets with k elements (for k a nonnegative integer) are order isomorphic, and therefore have the same order type (which is also an ordinal number). The ordinals for finite sets are denoted 0, 1, 2, 3, …, i.e., the integers one less than the corresponding nonnegative integers.

    The first transfinite ordinal, denoted omega, is the order type of the set of nonnegative integers (Dauben 1990, p. 152; Moore 1982, p. viii; Rubin 1967, pp. 86 and 177; Suppes 1972, p. 128). This is the “smallest” of Cantor’s transfinite numbers, defined to be the smallest ordinal number greater than the ordinal number of the whole numbers. Conway and Guy (1996) denote it with the notation omega={0,1,…|}.

    From the definition of ordinal comparison, it follows that the ordinal numbers are a well ordered set. In order of increasing size, the ordinal numbers are

    0, 1, 2, …, omega, omega+1, omega+2, …, omega+omega, omega+omega+1, ….

    The notation of ordinal numbers can be a bit counterintuitive, e.g., even though 1+omega=omega, omega+1>omega. The cardinal number of the set of countable ordinal numbers is denoted aleph_1 (aleph-1).

    Thus w is first transfinite ordinal, and all numbers feasibly reachable by +1 steps of succession from 0 will be finite and bounded by onward equally finite successors. Which is all that ordinary mathematical induction can reach.

    But the ellipsis of endlessness sets an impassable gulch to the far zone from w on.

    The tapes tell the tale:

    Pink, with near end:

    |0 === . . . k, k+1 ====> . . .

    Blue, pulled in k and cut just before k, continuing on to RHS endlessly:

    |k, k+1 ====> . . .

    Both can be set in 1:1 correspondence, and pull in k and match can be endlessly repeated beyond case o as above. (I just used do k times to get to k*k rows pulled in.)

    Where it is perhaps noteworthy that as Wolfram notes:

    The first transfinite ordinal, denoted omega, is the order type of the set of nonnegative integers (Dauben 1990, p. 152; Moore 1982, p. viii; Rubin 1967, pp. 86 and 177; Suppes 1972, p. 128). This is the “smallest” of Cantor’s transfinite numbers, defined to be the smallest ordinal number greater than the ordinal number of the whole numbers. Conway and Guy (1996) denote it with the notation omega={0,1,…|}.

    The bar at end of the ellipsis of endlessness is then highly significant, and we could write:

    {0, 1, 2, … |} –> omega, omega+1, omega+2, …, omega+omega, omega+omega+1, ….

    To parallel with:

    0, 1, 2, …, omega, omega+1, omega+2, …, omega+omega, omega+omega+1, ….

    Where the rows on the tapes now have a natural sense.

    Exploring further (I add: as the tapes are just that, continuous tapes with punched holes, so the issue of a continuum naturally arises . . . ), mild infinitesimal m –> 1/m –> A = w + g, beyond the span of the naturals and so also reals as usually conceived. Then m has neighbours such that descending from m to n we have 1/n –> B = w + (g +1), and as there is a continuum in [0,1] we can fit between m and n a range of mild infinitesimals that catapult to fill in the continuum between A and B.

    This is WLOG so it would make reasonable sense to suggest this as a continuum beyond the span of the naturals and reals, linked to infinitesimals close to 0 in the span [0,1], a continuum.

    Could this be a picture of the hyper real line emerging?

    (Though, I find the notion of a continuum [0,1] with infinitesimals near 0 that are effectively a gap of non reals, a bit troubling. I note, a practical definition is a number so small that m^2 ~ 0. That’s how Engineers, Applied Scientists and Physicists treat them. {10^-300}^2 = 10^-600 after all which vanishes in additions dominated by order 10^-300. Of course suitable steps have to be taken to yield a particular value and not just its closely near neighbourhood, hence limit approaches.)

    That would for me be a reasonable conjecture of coherent utility suitable for modelling type approaches as opposed to setting out on a chain of proof from a set of axioms. No claims for truth are made, only pragmatic utility tied to unification of a range of things that apart from unification are puzzling in a cognitive sense. A tentative so far explanation, not a proof. A suggestion, it looks like this might help make sense of, not a claimed proof.

    And, obviously such a view fits with the primitive, primary sense of the infinite seen in say the arrow pointing on forever on axes of a graph of the Cartesian plane or the Argand Plane.

    At this stage I feel a much lessened sense of concern, regarding what I have been looking at.

    KF

  513. KF,

    A, I must note to you that you are resorting to refusing a given answer stated in the explicit context that there is a gap in views that can make simplistic y/n without explanation meaningless.

    In the context of ZFC and standard definitions, there is no reason you cannot give a clear yes or no answer to Aleta’s question.

    Where, by definition of the tape, all along their length they have punched rows. Accordingly, it is hard to reject the point that there are spatially endlessly remote rows which duly have row-counts that are endlessly larger than any arbitrarily high but finite row value k. Which is the long way round to, these are infinitely far away. With appropriate row numbers.

    I’ve asked you to name a row which is infinitely far away from row 0. We have labels for each row (use any system of numerals you want). Can you do so?

    So now we come to labelling and giving values to rows in the tapes. For finite extension in steps that is not an issue. But the onward pointing to endlessness does raise an issue that is not resolved by saying ordinary induction takes them all in in one step. For, as the finite stepwise chain hits any k an endless succession always lies beyond. Bridging by an implicit sub axiom of pointing across an ellipsis of endlessness will be freighting that implicit step with the heavy lifting to reach the conclusion.

    Which leads to the point, why take that step?

    Are you asking why people generally accept the Axiom of Induction? If so, that’s an entirely separate issue from what I have been discussing, which is simply what theorems can be proved using this axiom.

    Concrete and readily represented, even turning glyphs into simple keyboard graphics:

    Pink, with near end:

    |0 === . . . k, k+1 ====> . . .

    Blue, pulled in k and cut just before k, continuing on to RHS endlessly:

    |k, k+1 ====> . . .

    Both can be set in 1:1 correspondence, and pull in k and match can be endlessly repeated beyond case o as above. (I just used do k times to get to k*k rows pulled in.)

    What mathematical language can we use to address that endlessness, resolving paradoxes?

    Well, the set of natural numbers is in 1-1 correspondence with the set of natural numbers k or greater, for any natural number k.

    Where’s the paradox?

    I think we’ve given adequate explanations of that fact in standard mathematical language.

    Have you found any sources which describe the same concerns you have, by the way?

    DS, BTW, a PA system vs chaining in sucession on Hilbert’s Hotel brings out the difference between all at once setwide processes and inherently finite chained successive ones. I don’t know about you but instantly on hearing of the hotel and what happens to provide room for fresh guests the issue is, how do you get the guests to all go to room 2n from room n? A PA system, with broadcast capability. By contrast with trying to propagate in a chain of steps.

    I think that particular detail is left unspecified. The room reassignment puzzle demonstrates a purely mathematical issue, and if you start worrying about logistics, you’re missing the point.

    For me, it first makes me a lot more wary of discussions of the infinite and the mathematics connected therewith. I have a lot of sympathy for the non standard analysis approach. Hyper reals and infinitesimals. I am now much more wary of discussions of naturals and of reals.

    Er, have you looked at the construction of the hyperreals? If you are wary of the natural and real numbers, you should be doubly so of the hyperreals. It’s a rather strange set.

    And of course, the natural numbers and reals are embedded in the hyperreals, so you are not going to get away from them in this way.

  514. KF,

    Thus w is first transfinite ordinal, and all numbers feasibly reachable by +1 steps of succession from 0 will be finite and bounded by onward equally finite successors. Which is all that ordinary mathematical induction can reach.

    That actually sounds more or less accurate. With ordinary mathematical induction, you prove statements such as “the sum of all positive integers at most n is n(n + 1)/2”. This applies only to natural numbers, each of which is finite.

    m –> 1/m –> A = w + g

    Ugh. You’re throwing around equations which have no solutions again. As I stated earlier in the discussion, this is like an assertion that a particular square is congruent to a particular circle.

    No claims for truth are made, only pragmatic utility tied to unification of a range of things that apart from unification are puzzling in a cognitive sense.

    ***

    At this stage I feel a much lessened sense of concern, regarding what I have been looking at.

    Well, if you are interested in the truth, you should still have a great deal of concern, as there are numerous errors in the above posts.

    If you really want to pursue this, I suggest reading up on the construction of the hyperreals (not an elementary textbook such as Keisler’s).

  515. PS to my #516:

    This appears to be a (relatively) accessible introduction to the hyperreals.

    I challenge you to read it and decide whether you are more comfortable working with the hyperreals vs. the natural and real numbers.

  516. DS,

    I thank you for sharing a link.

    However, I remain at a point where the concept of an actualised infinity of successive finite only values from 0 on in steps of +1 cannot seem reasonable.

    The largest finite values attained — being successively collections of prior sets in the succession (as was noted already in discussing axiom of infinity) — obviously would be inherently finite and will copy the sequence of counting sets so far. That was highlighted back in 217.

    So, it seems to me that an actualised collection of an infinite string of . . . finite value only . . . counting sets from 0 is not feasible. From what I see, the assertion that all that lurks under the ellipsis of endless continuation in {0,1, 2 . . . } will be finite will fail. How it fails is not that there are infinite attainable values, but that the transfinite zone cannot be operationally attained to. Ordinary induction chains and has the same counting embedded, as do the axioms. When we complete the domain conceptually by pointing across the ellipsis, we enfold a span that cannot be counted out. The all at once step, however, does not lead to oh all in the collection are finite. For the copy- of- the- set- so- far reason above.

    Instead, it seems to me that the ellipsis of endless continuation of counting sets and succession to such taken as a whole {0,1,2, . . . } lead rather to the transfinite order type w and onward as a concept, an idealisation beyond actual counting. That seems to resolve my concern, we only can operationalise finites but the succession may continue endless-LY and has order type w, cardinality aleph null.

    Then we may speak of the endless collection as a whole understanding we cannot operationally complete it.

    An endless tape, therefore obviously cannot be physically actualised no more than HH can, but the thought exercise is instructive; hence my remark that the tape records inspection reports on the HH rooms, one byte/row per room in succession.

    This points to the extended ordering of successive numbers from w on, beyond an impassable ellipsis of endlessness.

    That would be the remote far zone that could be contemplated, counting sets in succession cannot be extended operationally to a point of: “the next set k is now transfinite.”

    But endlessness — the key issue — has implications that can be used in an analysis, one of which is counting onwards from k, k+1 etc will 1:1 match the original order from 0, 1 etc, i.e. the set of endless successive rows is infinite and attempting to traverse the endless in steps will be futile; it will never attain to a transfinite value.

    Bringing in the ellipsis, {0,1,2 . . . } –> w, we are instantly at the zone of w on, which cannot be physically realised or attained to in a do forever loop but may profitably be mathematically discussed on an all at one go basis for {0,1,2 . . . }.

    So an endless continuation of tapes is conceptual and one may therefore reasonably suggest the far zone is the recognised transfinite one — in effect the tape dissolves into the extended ordered numbers.

    Where on this w etc would be beyond the naturals or reals. Resolving that concern-point. There is a qualitative difference thanks to the ellipsis of endlessness.

    The transfinite emerges from the endless-NESS of successive counting sets which can only be operationalised to finite extent. We cannot operationally complete an endless traverse of +/-1 steps.

    Which is the main point after all, and it is what brings a logical focus to the issue of a proposed endless causal succession as the past leading to the world of today.

    Such a proposed endless space-time past does not seem to be a tenable view. We cannot traverse an endless span in finite stage steps. (The issue of from k, k+1 on we can match the from 0,1 on shows how such will be frustrated. Endlessness is not realisable operationally in steps.)

    From that, we then may ask questions as to a unified far zone.

    KF

  517. w is not in N. Every number in N is finite. w is not in N, so it doesn’t negate the fact that all numbers within N are finite.

  518. PS: FWIW, Wiki on hyper reals:

    The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

    1 + 1 + . . . + 1. [–> so these will be transfinite],

    Such a number is infinite, and its reciprocal is infinitesimal. [–> hence the use of the catapult function y = 1/x to go between the neighbourhood of 0 and the relevant range] The term “hyper-real” was introduced by Edwin Hewitt in 1948.[1] . . . .

    The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form “for any number x…” that is true for the reals is also true for the hyperreals. [–> not for any x in R, x is finite] For example, the axiom that states “for any number x, x + 0 = x” still applies. The same is true for quantification over several numbers, e.g., “for any numbers x and y, xy = yx.” This ability to carry over statements from the reals to the hyperreals is called the transfer principle. However, statements of the form “for any set of numbers S …” may not carry over. [–> okay] The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties. The kinds of logical sentences that obey this restriction on quantification are referred to as statements in first-order logic.

    The transfer principle, however, doesn’t mean that R and *R have identical behavior. For instance, in *R there exists an element omega such that

    1 LT omega, 1+1 LT omega, 1+1+1 LT omega, 1+1+1+1 LT omega, . . . . [–> Obviously w is the first transfinite and is recognised as being in the hyper reals *R but not the reals R] [–> cf below, this w may be further along than the previous, i.e. there is a construction below that has predecessors and successors with surrounding ellipses of endlessness. That invites exploration but that is not primary for this thread’s purpose. Note my suggestion on mild infinitesimals above vs hard ones, on analogy of the catapulting function 1/x]

    but there is no such number in R. (In other words, *R is not Archimedean.) . . . .

    The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.

    The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Vladimir Kanovei and Shelah[4] shows that there is a definable, countably saturated (meaning omega-saturated, but not of course countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. —> in short just who are these guys is a debate but there is a suggestion as to how to unify] Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the continuum hypothesis . . .

    And on surreals

    In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. [–> I note my remaining qualms about the closed interval [0,1] and how it must be continuous] The surreals share many properties with the reals, including a total order LTEQ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. (Strictly speaking, the surreals are not a set, but a proper class.[1]) If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals.[2] It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories [–> norice, we are now in the zone of exploratory theories in Maths which work as explanatory models] it is not necessarily true that the surreals are the largest ordered field. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations . . . .

    In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: { | }. This representation, where L and R are both empty, is called 0. Subsequent stages yield forms like:

    { 0 | } = 1

    { 1 | } = 2

    { 2 | } = 3

    and

    { | 0 } = -1

    { | -1 } = -2

    { | -2 } = -3

    The integers are thus contained within the surreal numbers. (The above identities are definitions, in the sense that the right-hand side is a name for the left-hand side. That the names are actually appropriate will be evident when the arithmetic operations on surreal numbers are defined, as in the section below). Similarly, representations arise like:

    { 0 | 1 } = 1/2

    { 0 | 1/2 } = 1/4

    { 1/2 | 1 } = 3/4

    so that the dyadic rationals (rational numbers whose denominators are powers of 2) are contained within the surreal numbers.

    After an infinite number of stages [–> do forever loops again with ellipses of endlessness pointed across . . . ], infinite subsets become available, so that any real number a can be represented by { La | Ra } [–> reals imply do forever loops], where La is the set of all dyadic rationals less than a and Ra is the set of all dyadic rationals greater than a (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.

    But there are also representations like

    { 0, 1, 2, 3, … | } = omega

    { 0 | 1, 1/2, 1/4, 1/8, … } = epsilon

    where omega is a transfinite number greater than all integers and epsilon is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2*omega or omega – 1 and so forth.

    Looks like its a positive jungle out there full of stranger and stranger critters.

  519. Aleta, I have said w is not in N. What I have said is that the set we label N will have w as its order type and the ellipsis of endlessness is pivotal to its meaning. Where w does not appear out of nothing or sneak in magical properties, it is the order type resulting from endless succession that cannot be operationally completed stepwise. KF

    PS: Let me add, every number we can reach in N by a +1 stage sequence or extensions to that such as writing down in place value or scientific notation etc (all of which will involve series) will be finite and have another number beyond that will by iteration be finite. But due to endlessness we cannot exhaust N. Taking the +1 succession and the equivalent succession of sets, to go to actual endlessness — infeasible — would entail copying the set as a whole, endlessness within endlessness. But also this is the set we count by, if it has endless continuation there is an open endedness we CANNOT exhaust. That is where w steps in to point past the ellipsis.

  520. KF,

    1 LT omega, 1+1 LT omega, 1+1+1 LT omega, 1+1+1+1 LT omega, . . . . [–> Obviously w is the first transfinite and is recognised as being in the hyper reals *R but not the reals R]

    Eh? Do you have a reference?

    The problem is there is no “first transfinite” in the hyperreals.

    Given any positive infinite hyperinteger k, there exists infinitely many other positive infinite hyperintegers less than k.

    That means the ω above cannot be the same as the ordinal ω we have been referring to.

  521. #513 KF

    Rhetoric about denialism — which is horrifically loaded with Holocaust denial — does not help settle the matter. Nor does the notion, we do not dispute this, it is settled so there, only the ignorant, stupid, insane and/or wicked denialists would challenge the Consensus.

    EZ, I am looking at you here.

    You throw this into a purely academic discussion. YOU are loading the discussion. YOU are trying to make this into something else.

    YOU are poisoning the waters. Why, I don’t know. This discussion has nothing to do with Holocaust denial. So why bring it up? Why?

    And EZ, motive mongering or how I came up with the idea of pink and blue paper punch tapes is of little value. In fact, as Turing tapes of endlessness were discussed and as coloured 3+5 punched paper computer tapes exist, it seemed reasonable to look at parallel tapes running off endlessly and to ponder in ways that are anchored to an intuitive, simple entity. Thought exercises and abstracted models thereof are commonplace in several linked fields. And there is Hilbert’s hotel.

    And there is Hilbert’s hotel.

    Let me ask you this: have you had any of your work looked at by a group of reputable mathematicians? Have you submitted any of your ideas to any kind of magazine or peer-reviewed journal? Have you presented any of your ideas to a conference or gathering of bone fide mathematicians? I’m not trying to be rude I just want to know if you’ve HONESTLY had your contentions examined and critique by people who know the field.

  522. DS, there is a first transfinite ordinal w and there are hyper integers. The construction pointing to w seems reasonable to me as w being first ordinal greater than any chain of +1s from 0, interpreted on the von Neumann construction. Why, then, a call for specific reference on that? KF

  523. EZ, denialism — a term you tossed in like a live grenade — is so loaded a term that it is beyond the pale of civil discussion. Second, I have pointed out sufficient above that shows there is a jungle out there full of strange critters. And, the concept that we can tie hyper reals etc together is demonstrably on the table. It is also quite clear to me that ordinary induction is successive and that ever so many core points embed or imply do forever loops. Notice how a definition of reals progresses on what premise just above. So we do have to look at endlessness, which is indeed doing a lot of heavy lifting. The y = 1/x catapult does seem very fruitful. KF

  524. KF,

    DS, there is a first transfinite ordinal w and there are hyper integers.

    Yes, but there is no first transfinite positive hyperinteger.

    The ω in the hyperreal wikipedia article is not a von Neumann ordinal.

  525. Supporting Dave in 522:

    In the hyperreals

    There is infinitude of infinite integers, i.e., not finite elements of PA1?:

    … ? – 2, ? – 1, ?, ? + 1, ? + 2, …, 2?, 3?, …, ?² – ?, …, ?² – 1, ?², …

    which shows that our choice of omega was pretty much arbitrary. It’s more common to use the symbol omega instead. However, there is certainly a danger of confusing it with the first infinite ordinal number. The latter is of course defined uniquely.

    [Note: the Greek w shows u on the website where I have typed omega.]

    http://www.cut-the-knot.org/Wh.....bers.shtml

  526. Thanks for hunting that down, Aleta.

  527. DS, what is the least ordinal greater than any 1 +1 +1 . . . +1 where the string is not transfinitely long? KF

  528. Aleta, point taken, but I think the same question applies as was just posed to DS. KF

    PS: k as defined looks like enclosed by two ellipses of endlessness within the transfinites, i.e. it sits far out among transfinites.

  529. KF,

    DS, what is the least ordinal greater than any 1 +1 +1 . . . +1 where the string is not transfinitely long? KF

    The least ordinal greater than any 1 + 1 + … + 1 is ω.

    The least hyperinteger greater than any 1 + 1 + … + 1? There is no such thing.

    See Aleta’s link.

  530. Due to problems displaying Greek letters, here’s a correct 528:

    Supporting Dave in 522:

    In the hyperreals

    There is infinitude of infinite integers, i.e., not finite elements of PA1infinity:

    … k – 2, k – 1, k, k + 1, k + 2, …, 2 k, 3 k, …, k² – k, …, k² – 1, …

    which shows that our choice of k was pretty much arbitrary. It’s more common to use the symbol omega instead. However, there is certainly a danger of confusing it with the first infinite ordinal number. The latter is of course defined uniquely.

    [Note: a Greek K shows up on the website where I have typed k, and a Greek w where I’ve typed omega.]

    http://www.cut-the-knot.org/Wh.....bers.shtml

  531. DS, I just noted, there are two ellipses of endlessness surrounding the transfinite k and neighbours, which looks interestingly inviting of a siting suitably remote from w among transfinites. But this is just a look not an argument. My real point is there is a jungle out there that has strange critters of unknown linkages inviting some unification. On that front try: http://shelah.logic.at/files/825.pdf on candidates to be “the” — even that is controversial it seems — hyper reals. KF

  532. kf writes,

    PS: k as defined looks like enclosed by two ellipses of endlessness within the transfinites, i.e. it sits far out among transfinites.

    That’s a strange thing to say. We could write the integers as

    {… -3, -2, -1, 0, 1, 2, 3, …} 0 sits between two ellipses: Is it “far out” among the integers? Is it more or less far out than 70 billion billion billion?

  533. Aleta,

    pardon but k is transfinite, o is not. I suggest that makes a difference for looking at them.

    I notice as follows as a simple clip at Wolfram:

    Transfinite Number

    One of Cantor’s ordinal numbers omega, omega+1, omega+2, …, omega+omega, omega+omega+1, … which is “larger” than any whole number.

    which is suggestive.

    I essay no proof-claim there, I just notice a resemblance that would “fit” with a hard infinitesimal catapulting to something in say the bolded range or one of its many onward cousins.

    But what I am really saying is, how does the jungle get set in order with a tree of unifying relationships?

    Particularly among the trans-finites?

    Surreals? Whatever?

    This being beyond main focus but relevant to the principle that knowledge claims should be unified where possible.

    GEM

  534. KF,

    But what I am really saying is, how does the jungle get set in order with a tree of unifying relationships?

    Particularly among the trans-finites?

    Surreals? Whatever?

    I don’t know anything about this, but last post on this page at stackexchange discusses the relationship between hyperreals and surreal numbers, with a link to this paper. Theorem 1 states:

    Whereas R is (up to isomorphism) the unique homogeneous universal Archimedean ordered field, No [the surreal numbers] is (up to isomorphism) the unique homogeneous universal ordered field.

    The precise definitions are given in the paper, but the key one is (paraphrased):

    An ordered field A is said to be universal if every ordered field whose universe is a class of NBG can be embedded in A.

    Apparently every ordered field can be embedded in the surreal numbers, so it’s “large as possible” in some sense. This is based on the NBG axiom system for set theory rather than ZFC.

  535. DS,

    That does look promising, Abstract:

    Abstract.

    In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including [lists] to name only a few. Indeed, this particular real-closed field, which Conway calls No , is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG (von Neumann–Bernays–Godel set theory with global choice), it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system R of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas R should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), No may be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal num-bers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-a-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum.

    KF

  536. PS: Footnoted to OP is a picture of Ehrlich’s grand number tree. This sets a context for discussion in a unified context with room enough for pondering numbers great and small through the surreals. Not to mention rather unusual operations on surprising numbers.

  537. F/N: Headlined (HT DS), the numbers sandbox is now officially open for play. KF

  538. F/N: Wiki has an interesting remark on ordinary mathematical induction:

    Having proven the base case and the inductive step, then the structure of {N} is such that any value can be obtained by performing the inductive step repeatedly. It may be helpful to think of the domino effect. Consider a half line of dominoes each standing on end, and extending infinitely to the right (see picture) [–> similar to the punched tape]. Suppose that:

    The first domino falls right.

    If a (fixed but arbitrary) domino falls right, then its next neighbor also falls right.

    With these assumptions one can conclude (using mathematical induction) that all of the dominoes will fall right. [–> strictly, that the dominoes in sequence from the first will be subjected to a chaining process propagating stepwise from one to the next, however this then runs into endlessness of the chain as set up]

    If the dominoes are arranged in another way, this conclusion needn’t hold (see Peano axioms#Formulation for a counter example). Similarly, the induction axiom describes an essential property of {N}, viz. that each of its members can be reached from 0 by sufficiently often adding 1 [–> do forever looping creating stepwise succession]

    That is, we see the do forever iteration implicated in the process. This runs into the problem that for any k achieved in k finite steps, from k, k+1 on is just as much able to be put in 1:1 correspondence with the undisturbed set as before.

    This illustrates a case in point of pointing across an ellipsis of endlessness.

    KF

  539. KF,

    That’s the misleading aspect of the domino model I referred to somewhere above.

    It implies that mathematical induction is simply the application of the law of detachment over and over, which is not the case.

    Without the Axiom of Induction, you would be right.

    But this axiom does the “heavy lifting” that you refer to, omitting the do forever loop.

    It in essence says you can replace these “do forever” loops with one (or just a few) steps.

    Edit:

    the induction axiom describes an essential property of {N}, viz. that each of its members can be reached from 0 by sufficiently often adding 1 [–> do forever looping creating stepwise succession]

    This simply states that any natural number is the k-th successor to 0, for some finite k. It doesn’t say anything about do forever loops, and that’s not how N is constructed.

  540. F/N: Ehrlich, pp 7 – 8:

    Among the striking s-hierarchical features of No is that much as the surreal numbers emerge from the empty set of surreal numbers by means of | a transfinite recursion that provides an unfolding of the entire spectrum of numbers great and small (modulo the aforementioned provisos), the recursive process of defining No’s arithmetic in turn provides an unfolding of the entire spectrum of ordered fields in such a way that an isomorphic copy of every such system either emerges as an initial subtree of No or is contained in a theoretically distinguished instance of such a system that does.

    KF

    PS: kth successor –> successive what . . . ?

  541. KF,

    PS: kth successor –> successive what . . . ?

    Natural number?

    You can construct any finite set {0, 1, 2, …, k} using the successor operation.

    To construct the set N requires the Axiom of Infinity.

  542. F/N: Ehrlich, with emphasis:

    >>Among the striking s-hierarchical features of No is that much as the surreal numbers emerge from the empty set of surreal numbers by means of | a transfinite recursion that provides an unfolding of the entire spectrum of numbers great and small (modulo the aforementioned provisos), the recursive process of defining No’s arithmetic in turn provides an unfolding of the entire spectrum of ordered fields in such a way that an isomorphic copy of every such system either emerges as an initial subtree of No or is contained in a theoretically distinguished instance of such a system that does.>>

    Recursion and unfolding all the way up and down.

    Where AmHD:

    re·cur·sion (r?-kûr?zh?n)
    n.
    1. Mathematics
    a. A method of defining a sequence of objects, such as an expression, function, or set, where some number of initial objects are given and each successive object is defined in terms of the preceding objects. The Fibonacci sequence is defined by recursion.
    b. A set of objects so defined.
    c. A rule describing the relation between an object in a recursive sequence in terms of the preceding objects.

    In short a pattern is established then it is universalised by pointing across ellipsis of endlessness. Which is also embedded in relevant axioms.

    KF

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