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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.

Comments
KF,
DS, we patently cannot traverse an endless span of steps of warrant in either direction step by step, pivoting on our finitude and fallibility given that a chunk of time, effort, energy will be taken up by each successive step in building a worldviws case. Pointing across the ellipsis of endlessness and imposing a conclusion is a finitely remote final step, even in Mathematics. And this ‘ent Maths, it is warranting worldviews. Why the persistent sidetrack on a tangential matter when you have already in effect acknowledged tangentiality? KF
Well, when you state "the infinite regress is absurd, we cannot even get to A step by step from infinity”, you are are certainly making a mathematical statement. This "warranted worldviews" argument makes a much weaker claim than the above, however. You are not showing that infinite regresses are absurd, or even that they raise any logical problems. Rather you are arguing that finite humans cannot comprehend them. The upshot is that we cannot dismiss the possibility that infinite regresses exist, which then leaves a gap in your argument.daveS
June 2, 2016
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KF, Interesting proof published earlier this year: Mathematicians Bridge Finite-Infinite Divide
With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary. The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist. Mapping and understanding this division is “at the heart of mathematical logic,” said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality.
The actual paper.daveS
May 25, 2016
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KF,
The claim is, one is coming from the transfinitely remote, thus across a zone of endlessness.
Well, no, the ladder has infinitely many rungs, each finitely many steps from the ground. I've made no claims about coming from the "transfinitely remote".
Now, if our man was descending without beginning but at no time was he ever endlessly remote in stages, he was always only finitely removed.
Yes, that's true. I think we're making progress.
That is, his past becomes a problem: finite remove and finite stages will not span endlessness.
The fact that he was always finitely many steps from the ground is completely consistent with the fact that he has descended every one of the infinite number of steps on the ladder. n seconds ago he was on rung number n, for all natural numbers n. No rungs were missed.
What I think you are saying is in effect we have had a potentially infinite past. But that is just what we cannot have had. The real past, whatever it was, HAS to have been actualised. It is how we got here.
I am saying the past comprises an actual infinity (in this thought experiment).
The future is different, we may successively and open-endedly add, pointing onward.
Yes, just as ω and ω* are different.
We can specify a set with endlessness in it and may conceptually deliver it in an all at once grand step (which typically points across an ellipsis of endlessness) but what we never do is actually traverse the span in successive, cumulative finite stage steps.
What was actually traversed in successive finite stages was accumulatively finite. Must be. Finite past is implied.
Let's review:
1. The ladder has infinitely many rungs (in 1-1 correspondence with the natural numbers). 2. The man was on rung n, n seconds ago, for every natural number n. 3. The man has just reached rung 0.
What we have is a man descending an infinite ladder throughout an infinite past, just now completing his traversal of every rung.daveS
May 6, 2016
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DS, You will see that I pointed to the case of a start-point then the case of beginningless descent on stipulation of an endlessly remote zone. (This points to the tree of surreal numbers, cf Ehrlich in OP as augmented.) The claim is, one is coming from the transfinitely remote, thus across a zone of endlessness. In particular, it is claimed or implied that our cosmos -- in relevant causally successive stages -- has come from an endlessly remote past to the present. Descending a ladder that is endlessly high to reach its foot is a drawing out of the essence of that. Now, if our man was descending without beginning but at no time was he ever endlessly remote in stages, he was always only finitely removed. That is, his past becomes a problem: finite remove and finite stages will not span endlessness. What I think you are saying is in effect we have had a potentially infinite past. But that is just what we cannot have had. The real past, whatever it was, HAS to have been actualised. It is how we got here. The future is different, we may successively and open-endedly add, pointing onward. In both cases, what we never actually traverse in finite stage steps is an endless span, such as is represented by an ellipsis of endlessness. The pink.blue tape example shows us why. It is in the endlessness that the transfinite property lies. We can specify a set with endlessness in it and may conceptually deliver it in an all at once grand step (which typically points across an ellipsis of endlessness) but what we never do is actually traverse the span in successive, cumulative finite stage steps. That seems firmly shown. And that is why the requirement of actual traversal in causally successive stages of the real past . . . whatever it was . . . to achieve the present (comparable to the successive accumulation of stepping up or down a ladder) implies finitude. What was actually traversed in successive finite stages was accumulatively finite. Must be. Finite past is implied. There was a distinct beginning to our causally successive world. This is also a clue to root of our world. There is now necessitated an a-causal, non successive finite stage accumulated root for our world. As I have said, non-being has no causal powers, and were there ever utter nothing, there thus would have been no succession -- nothingness would forever obtain. A real cosmos from true nothing is impossible. A cosmos now is and is of successive causal stage character, pointing to a beginning. The root of that beginning is necessary being that is of diverse character, no temporally successive causally dependent accumulation . . . or else we are just regressing the problem. We are looking into the strange conceptual world of eternal, necessary being as root of any temporal, causally successive world. The image that comes to mind is of lines of longitude converging at a north pole that is simultaneously due north of them all. And if this is beginning to sound like a scrunching of the spacetime domain into a point at infinity in effect that swallows them all, or as pointing to a higher order hyper space world or even like "in him we live and move and have our being" that is what it looks like we are peering into. The world of eternal forms is taking its revenge. Eternally contemplated in eternal mind. Mind with power to spin out a causally successive world: fiat lux . . . ! KFkairosfocus
May 5, 2016
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KF,
In various ways, this has been discussed many times.
That's definitely an understatement! :-)
By the nature of this situation if one has been descending and reaches down to 0, his start point was finitely not endlessly remote.
Of course in the ladder example, it is assumed there was no start point.
And if he never started but was always descending in finite stage finite time steps, so long as he was actually endlessly remote he cannot traverse the endless span to reach a finite near neighbourhood of the 0 point.
He never started, but he also was never "endlessly remote" from the bottom rung. Do we have that squared away now?
1) There is/was no starting point. 2) At no time was he infinitely far from the bottom rung.
daveS
May 5, 2016
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DS, the issue is the nature of endlessness. Once there is an endless span, as the pink vs blue tape example shows, on reaching any large but finite k, the k, k+1, k+2 etc can be placed in 1:1 correspondence with 0,1, 2 etc. This is a part of defining the counting numbers as infinite and two sets in 1:1 match will be of the same cardinality. That is, endlessness dominates and both are infinitely large, cardinality aleph null. So, going up stepwise to the far zone one cannot complete traversal of the endless. Coming down, the same endless span confronts the one who would come down from the endlessly remote to the point where the ladder stands on the ground, its level 0. By the nature of this situation if one has been descending and reaches down to 0, his start point was finitely not endlessly remote. And if he never started but was always descending in finite stage finite time steps, so long as he was actually endlessly remote he cannot traverse the endless span to reach a finite near neighbourhood of the 0 point. In various ways, this has been discussed many times. The pivotal issue is traversal of the endless span. KFkairosfocus
May 5, 2016
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KF,
And it should be clear that if endlessness cannot be spanned going up, it cannot be spanned in steps down too.
That's exactly the question; and no, it isn't clear to me and to a number of authors who have published on this subject. What exactly is the justification? I would think that if this is true, you should be able to demonstrate it with no mention whatsoever of climbing up the ladder.daveS
May 5, 2016
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DS Yes we are quite relieved at relative good news. The logic is direct, once distinct identity applies all three do. A sandbox where LEM may not, is a case where distinct identity is needed to get there. The problem with counting up to/down from the span of the endless ladder lies in the endlessness. Once that is there ahead, at any finite k, k, k+1 etc can go in 1:1 match endlessly with 0,1,2 etc. So you never get beyond a finite span in steps, endlessness remains before you. And it should be clear that if endlessness cannot be spanned going up, it cannot be spanned in steps down too. KFkairosfocus
May 5, 2016
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KF,
(And a diagnosis on a close person has been dialled back, thank God.)
I'm glad to hear that. If you do run across a published argument showing that LEM and LNC are corollaries of LOI, then I would appreciate you posting it. Now regarding the ladder scenario:
Ponder such a ladder and imagine one of those inking counters. Have someone stamp a bottom plate 0, then rungs 1,2, 3 . . . but, you cannot complete stamping going up. Now imagine said ladder is being stamped by someone else counting down. He steps off rung 1, bends over on his knees (which pop alarmingly) then brushes away a very long beard and stamps 0 on the plate, gets up and says, finished. This cannot be done counting down, no more than it can be done going up. For the same reason of attempted completing of endlessness in steps.
Under the assumptions I am making about time, I agree that the person climbing up the ladder cannot complete the job in time. That's because I am assuming that every moment in the future is finitely many seconds from the present (or from the time he began climbing). But why is it impossible for the person climbing down to finish inking all the rungs? I don't see anything other than an assertion here.daveS
May 5, 2016
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PS: While I am now here, the past week or so has seen an all quiet for the moment including a cluster of bad dogs that suddenly stopped barking after a major public meeting fizzled, with a sitting following being distinctly tame. (And a diagnosis on a close person has been dialled back, thank God.) I think my mind can give enough focus here for the moment, to comment a little bit on infinite successions in steps. My suggestion is that the relevant order type in a ladder of endless succession of rungs is w, omega. Ponder such a ladder and imagine one of those inking counters. Have someone stamp a bottom plate 0, then rungs 1,2, 3 . . . but, you cannot complete stamping going up. Now imagine said ladder is being stamped by someone else counting down. He steps off rung 1, bends over on his knees (which pop alarmingly) then brushes away a very long beard and stamps 0 on the plate, gets up and says, finished. This cannot be done counting down, no more than it can be done going up. For the same reason of attempted completing of endlessness in steps.kairosfocus
May 5, 2016
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DS, My concern is not logics but the world in which such logics can be conceived. The sandbox logic should be understood to be just that -- a sandbox in a world. This is similar to the presence of Observer in Quantum work. Yes, we can set up a sandbox algebra as you describe but to do so we have to use distinct symbols or concepts which means we are relying on the triple laws. KFkairosfocus
May 5, 2016
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KF,
DS, to get to fuzzy logic, you have to use things that have distinct identity starting with alphanumeric characters. The same obtains for quantum theory. Refining a partial set member concept requires thought using distinct identity and its immediately present co-laws. Try to conceptualise and communicate or reason without marking distinctions — impossible. So, the way we think about fuzzy logic has to reckon with that or fall into incoherence. KF
But I'm not saying anything contrary to distinct identity. In fact, I _believe_ there are fuzzy logics with two-valued identity relations but many-valued predicates. Therefore LOI is true but LEM (could be) false in such logics.daveS
May 5, 2016
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KF,
I remain however quite busy here with developments so I cannot give much attention.
That's fine, no rush. With some bolding added:
I do note that if a stepwise finite stage traversal of the endless is impossible then it follows that this cannot be accomplished — completed, thus ending the endless (language here is trying to warn us) — in stages from the past to the present as much as from the present to the future.
Again, this hasn't been shown. That we cannot traverse a set of order type ω in time does not imply that we cannot traverse a set of order type ω* in time. You have not addressed the asymmetry between the two order types. If you believe that traversing sets of order type ω* in time is impossible, then you should be able to prove such without ever referring to any sets of order type ω.daveS
April 19, 2016
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F/N: Putting background for the renewal of this long lived thread: https://uncommondescent.com/darwinism/pastafarians-not-giving-up-their-claim-to-be-a-religion/#comment-603347 >>37 daveSApril 19, 2016 at 6:22 am KF, Infinite regress being unattainable as endlessness cannot be traversed in finite stage steps, *Proof pending? 39 kairosfocusApril 19, 2016 at 6:34 am DS, proof given over and over, as the case of the thought exercise of pink and blue punched tapes shows. start both at 0, advance blue to arbitrarily large but finite k, then k+1 etc. Put blue from k in complete, endless 1:1 match with pink from 0. This shows pink is infinite and blue from k on is the same. Moreover as endlessness onward is always there from any finite k, no process of repeated finite stage steps can traverse that endlessness. I suggest this side issue goes back to its proper thread. KF 40 daveSApril 19, 2016 at 6:36 am KF, Thanks, I’ll respond in the other thread.>> I remain however quite busy here with developments so I cannot give much attention. I do note that if a stepwise finite stage traversal of the endless is impossible then it follows that this cannot be accomplished -- completed, thus ending the endless (language here is trying to warn us) -- in stages from the past to the present as much as from the present to the future. Infinite sets are delivered all at once, often by giving a general case and pointing to endlessness and across it. From potential to actual. But at no point is there ever an actual completion of endlessness in steps. with the tapes, for any k in our k-register, there will be always endlessness onwards from k+1 etc an infinite set is always there in front, untraversed from any finite k. KFkairosfocus
April 19, 2016
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KF,
DS, proof given over and over, as the case of the thought exercise of pink and blue punched tapes shows. start both at 0, advance blue to arbitrarily large but finite k, then k+1 etc. Put blue from k in complete, endless 1:1 match with pink from 0. This shows pink is infinite and blue from k on is the same. Moreover as endlessness onward is always there from any finite k, no process of repeated finite stage steps can traverse that endlessness. I suggest this side issue goes back to its proper thread. KF
Modulo some issues with vocabulary, I don't disagree with this. Here's what I was referring to, with emphasis added:
Infinite regress being unattainable as endlessness cannot be traversed in finite stage steps,
I take this to mean that (paraphrasing) "because we cannot construct N starting with 0 and the successor operation, an infinite regress (for example, an infinite past) is impossible" . That is what I'm objecting to. The tapes example simply shows that two sets are infinite. It doesn't show that an infinite past (or an infinite regress) is impossible.daveS
April 19, 2016
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KF,
I pointed to Godel’s theorems regarding complex systems in Maths.
Well, the real numbers are reasonably complex. Furthermore, I think much of our discussion could be conducted in the theory of Presburger arithmetic, which is also known to be complete and consistent. I'm concerned that we don't use Gödel's theorems to cast doubt on all forms of axiomatic reasoning, even when they don't apply.
Potential vs actually completed infinity is a relevant concept.
Maybe, but I'm not aware of it appearing in mathematics.
I suggest pondering HeKS’ ladder (recall Russell’s village barber illustration) to see the challenge of completed infinite descent. KF
I haven't seen any straightforward contradiction derived from the ladder illustration, as there is with the barber paradox. I still stand by my claim that none of the arguments presented so far against an infinite past succeed.daveS
April 9, 2016
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DS, again, my focus has to be local. I pointed to Godel's theorems regarding complex systems in Maths. Potential vs actually completed infinity is a relevant concept. I suggest pondering HeKS' ladder (recall Russell's village barber illustration) to see the challenge of completed infinite descent. KFkairosfocus
April 9, 2016
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KF,
Including that post Godel, axiomatic systems are incomplete or incoherent, and there is no constructive process that guarantees coherence of axiom systems.
To the contrary, there are many axiomatic systems that are known to be complete and consistent. For example, the theory of real numbers is both complete and consistent.
This means succession to traverse the set from 0 cannot be completed and that ordinary mathematical induction addresses the potential not the completed infinite.
I think we are all in agreement that constructing N starting from 0 in "+ 1" steps as you say, is not possible. I'm not clear why you are repeating this once more. Furthermore, I'm not aware of the distinction between "potential" and "complete" infinities being made in the mathematical systems we have been discussing. Sets are either infinite or not. If you want to explore further, I suggest considering the traversal of sets of order type ω* in order as a model of an infinite past. That's what I've been talking about all along, but I believe you have primarily focused on sets of order type ω, which are different.daveS
April 9, 2016
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PS: Alternative frames for set theory: http://plato.stanford.edu/entries/settheory-alternative/ also a note on the number tree: https://uncommondescent.com/mathematics/fyi-ftr-on-ehrlichs-unified-overview-of-numbers-great-and-small-ht-ds/ as well as, the surreals (HT DS): http://www.ohio.edu/people/ehrlich/Unification.pdfkairosfocus
April 9, 2016
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Folks, I think that -- absent some good fresh Mathematics and/or phil of Math put on the table, this thread will make little further progress. Where also, local events are necessarily drawing my focus in a way that means I cannot do justice to discussions here for a while at least. In that light, I am thinking the best I can do just now is suggest that endlessness is a key to understanding infinity and the surreals seem to have a lot of promise on that. I am led to conclude that the oxymoronic phrase, trying to end the endless by traversing it in finite stage steps, is not only a matter of a form of words. It speaks to core realitues and constraints involving coherence, logic and the logical analysis of structure and quantity, aka mathematics. This includes causally connected stages comparable to descending an infinite ladder to reaching just now arriving on the ground. Such can be extended to cosmological unfolding and makes claims of an infinite past quasi-physical causal stage succession cosmos dubious. BTW, that is where I think Mathematics gains its relevance and causal influence, the coherence of distinct identity and the need that all things present in our world be sufficiently mutually compatible that they can exist together in a common actualised domain or possible world. Mathematics is not magic off by itself. However, that means it becomes foundational to understanding reality, and that we must bear in mind its strengths and limitations. Including that post Godel, axiomatic systems are incomplete or incoherent, and there is no constructive process that guarantees coherence of axiom systems. Mathematics is not a stand-in for absolute unquestionable truth established by circles of adepts. This too becomes an open ended, hopefully progressive research programme. I remain of the view that the ellipsis of endlessness is key to understanding the successive counting numbers. Any given k has k+1 etc in succession that can be 1:1 matched to the full set from 0,1,2 on. That is the original set and the set that is bijective with it are both infinite. This is an illustration of how endlessness and infinity are connected. This means succession to traverse the set from 0 cannot be completed and that ordinary mathematical induction addresses the potential not the completed infinite. Transfinite induction goes beyond that but requires a more stringent process, outlined fairly recently above. Also, this leads me to the view that all counting sets we can reach to specifically in +1 steps, or represent as numerals based on such (place value, sci notation etc) or symbolise specifically will be finite but we must reckon with onward endlessness lest we draw conclusions that can get us into trouble. I for now hold to the view that the conclusion on ordinary math induction that all counting numbers constitute an infinite sequence of only finite values is a step too far, failing to adequately reckon with onward endlessness. I would rather say, what we can reach is finite but the endlessness onward implies we cannot fully define its members. Instead we recognise endlessness as a new quantitative phenomenon and assign the numeral omega, w for convenience above. On to the surreals for exploring the jungle of numbers great and small. Yes, a minority view, but I cannot honestly go beyond what I see in the logic at any time; even if I have to say simply, I do not follow this could you show me the steps I may have missed. I of course remain open to further understanding. As a closing word, thanks for a civil, generally serious discussion. KFkairosfocus
April 9, 2016
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KF,
Ds, passing by. I say the opposite, we CANNOT complete an actual infinity in finite stage steps.
I'm agreeing with that above, at least in a mathematical context. ω cannot be "completed" in finite stage steps, that is, by starting with {} and applying the successor operation repeatedly.
And the infinite past issue has to do with exactly that succession of finite stages. KF
Well, the connection is unclear to me. First of all, an infinite past is not a deduction or construction in first-order logic. Second, an infinite past would have no beginning, whereas a deduction or construction in first-order logic would.daveS
April 8, 2016
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at 1392, ellazimm quotes Wikipedia as saying
The set of finite ordinals is infinite, the smallest infinite ordinal is omega (spelled out because WordPress didn’t take the Greek letter)..
This seems correct to me. Then kf says,
Wiki should take a pause but likely won’t. The value w is not as a succession to definite finite steps say k –> k+1 –> w, but instead it is a recognition of their endlessness as a distinct quantitative phenomenon worth recognising with a symbol.
This second part seems correct to me: "w is a recognition of their endlessness it is a recognition of their endlessness as a distinct quantitative phenomenon worth recognising with a symbol.with a symbol" but I don't think that the Wikepedia quote said that "the value w is a succession to definite finite steps". The Wikipedia quote recognizes that the set of finite ordinals is infinite, and the kf says what I have said a few times: that a new number and symbol were created by Cantor as a name for this "distinct quantitative phenomenon."Aleta
April 8, 2016
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Ds, passing by. I say the opposite, we CANNOT complete an actual infinity in finite stage steps. Cf what happens onward when one reaches ant k. And the infinite past issue has to do with exactly that succession of finite stages. KF PS: EZ, Wiki should take a pause but likely won't. The value w is not as a succession to definite finite steps say k --> k+1 --> w, but instead it is a recognition of their endlessness as a distinct quantitative phenomenon worth recognising with a symbol.kairosfocus
April 8, 2016
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KF,
KF:
Namely, we are dealing with the capital case divergent sequence, it is HOW we count up without limit. And by definition as we go it expands in value endlessly. So, on the copy the set so far principle to get to the next value, if endlessness WERE completed, it would entail endless members of the set of counting sets. That is, the claim, an infinite succession of finite incrementing values is not valid as were it completed not all values would be finite.
Me:
Can you prove this? I have interpreted this as saying that ω has elements (sets) of infinite cardinality. If that’s what you mean, then this statement is false. You are of course invited to prove this statement if you disagree.
PS to this exchange: Another interpretation of your statement is that you are speaking of starting with {} and applying the successor operation infinitely many times. But as I have stated, this construction is simply undefined in first-order logic so it doesn't tell us anything about ω, the set of counting sets.daveS
April 8, 2016
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KF #1390 From the Wikipedia page on ordinal numbers:
The set of finite ordinals is infinite, the smallest infinite ordinal is omega (spelled out because WordPress didn't take the Greek letter)..
Apparently Wikipedia doesn't have a problem with an infinite set of finite values either.ellazimm
April 8, 2016
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KF,
This brings back a main point I have made all along, we cannot complete endlessness in finite stage steps such as +1 counting steps. (And yes that goes to the thread’s main point on claimed infinite past causal stage succession to get to our now world.)
I would express your first sentence as: you cannot construct N starting with {} and the successor operation in first-order logic. That certainly doesn't prove that an infinite past is impossible.
Namely, we are dealing with the capital case divergent sequence, it is HOW we count up without limit. And by definition as we go it expands in value endlessly. So, on the copy the set so far principle to get to the next value, if endlessness WERE completed, it would entail endless members of the set of counting sets. That is, the claim, an infinite succession of finite incrementing values is not valid as were it completed not all values would be finite.
Can you prove this? I have interpreted this as saying that ω has elements (sets) of infinite cardinality. If that's what you mean, then this statement is false. You are of course invited to prove this statement if you disagree.
We may proceed to recognise a new quantity, w, for order type of the endless succession.
That's where the Axiom of Infinity comes in. We (most of us, anyway) postulate that we have a set ω that has {} as an element and which is closed under the successor operation. So again, ω is not "completed" through a series of + 1 steps.
That is very different from ending the endless in finite stage steps.
Yes, exactly. We do not "end the endless" through finite stage steps, since we can't do that in first-order logic. The existence of the completed set ω is assumed.
Also, ordinary mathematical induction on ase 0 then case k => case k+1 implies a chained succession, whether we want to look at the chain as a whole or at its linking successive property or generalise on that. The very k, k+1 chaining points to the same onward endlessness issue. So, in a case like this, due caution is needed when endlessness has material force.
Hmm. If you're not saying that these inductive proofs have infinitely many steps, I have no argument here. If you are saying this, I would again point out such proofs are impossible.
Yes, we can set up axioms etc, but we must not forget their potential for error or incoherence, as Godel highlighted. Nor should we forget that when an imposed premise p is directly responsible for the force of a conclusion C, the locus shifts to, why that P?
Well, there's always the possibility of error or inconsistency. How else can we do mathematics other than by setting up axioms and logical systems? If we don't have a "rule book", so to speak, then how do we determine what is allowed?daveS
April 8, 2016
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EZ, I pass by briefly. I am using w for OMEGA (an ordinal) not aleph-null, which is in effect the cardinality of the first range of transfinite ordinals w, w+1 etc. The cardinality of the reals is an irrelevance to that. I have no problem with diagonalisation. And I suggest that you may find it helpful to look at the clips on surreals added to the OP long since [HT DS], which show a tree of numbers great and small. This seems to give a good framework for addressing the overall structure of quantity, complete with many ellipses of endlessness. KFkairosfocus
April 8, 2016
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KF #1388 Since I don't think you've said anything new here and I don't have anything new to say either it seems pointless to pick my way through your statement. I did read it all. You are using a small omega to stand for what is normally called aleph-null, the smallest infinite cardinal number, the cardinality of the integers (and the rational numbers). Any set with that cardinality is said to be countably infinite. Cantor proved that the cardinality of the real numbers is greater than that of the rationals. Do you agree with his proof? Are there more real numbers than rational numbers? Just curious. You seem to just want to lump all infinite sets into 'endlessness'.ellazimm
April 8, 2016
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Folks, As noted, local issues are peaking. That said, a couple of quick notes. First, mathematics is far more than axiomatised systems, though those are important in pulling together fields of thought. Next, it is clear the successive counting sets from 0 are just that as von Neumann's construction shows. This requires that a coherent view brings that to bear and answers to it. In that context, patently once we have arbitrarily large but finite value/set of counting sets so far k, k+1 etc beckon onwards. Indeed that is how we know k is finite, it is bounded on the upper side. But as k, k+1 etc can be matched endlessly with 0,1,2 etc, this means BOTH are infinite [0,1,2 etc by way of matching with a proper subset, k on as it matches with a set just shown infinite . . . a bit of a subtle difference from mere circularity], though to a certain extent that is repeating, endlessness is the essence of being infinite. This brings back a main point I have made all along, we cannot complete endlessness in finite stage steps such as +1 counting steps. (And yes that goes to the thread's main point on claimed infinite past causal stage succession to get to our now world.) It is also the pivot of a point that does not depend for its force on whether there is abundant or even sparse discussion of it in the literature. Though, the finitists of today have a few pungent observations to make. Namely, we are dealing with the capital case divergent sequence, it is HOW we count up without limit. And by definition as we go it expands in value endlessly. So, on the copy the set so far principle to get to the next value, if endlessness WERE completed, it would entail endless members of the set of counting sets. That is, the claim, an infinite succession of finite incrementing values is not valid as were it completed not all values would be finite. But as was reminded of, endlessness cannot be completed in such +1 steps or the like. That is we have a potential infinity on this line, and it is relevant to highlight the ellipsis of endlessness onward. We may proceed to recognise a new quantity, w, for order type of the endless succession. That is very different from ending the endless in finite stage steps. A safer conclusion, then, seems to be that every particular value we can succeed to or represent, k, will be finite but endlessness continues onward and cannot be captured by using specific values. Hence the ellipsis of endlessness. Also, ordinary mathematical induction on ase 0 then case k => case k+1 implies a chained succession, whether we want to look at the chain as a whole or at its linking successive property or generalise on that. The very k, k+1 chaining points to the same onward endlessness issue. So, in a case like this, due caution is needed when endlessness has material force. In such cases we may be well advised to look at transfinite induction (including, I just point to, how it handles limit ordinals such as w). Yes, we can set up axioms etc, but we must not forget their potential for error or incoherence, as Godel highlighted. Nor should we forget that when an imposed premise p is directly responsible for the force of a conclusion C, the locus shifts to, why that P? We could go on, but that is enough for now on balance. Again, sorry, local pressures. KFkairosfocus
April 8, 2016
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Pardon, local issues are peaking again.kairosfocus
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