In the current UD thread on Darwinism and an infinite past, there has been an exchange on Spitzer’s argument that it is impossible to traverse an infinite past to arrive at the present.

Let me share and headline what is in effect the current state of play:

DS, 108: >>KF,

DS, ticking clocks meet dying stars and death of cosmos as useful concentrations of energy die out.

There are oscillating universe models which are consistent with an infinite past, as I stated. Replace each tick with a big bang/crunch cycle.

And that an actually transfinite number of ticks can in principle occur is the precise thing to be shown.

No. I am saying that Spitzer assumes that an infinite number of ticks *cannot* in principle occur as part of his argument. The burden is on him to prove that.

A down to 2, 1, 0. Where A = 1/m, m –> 0 i.e. is infinitesimal. KF

There are no infinitesimals in sight in your statement above. All the numbers are real and finite.

Now that we’ve clarified the ticking vs. counting issue, do you still have mathematical (not physical) objections to the eternal ticking clock example? If so, I challenge you to take some time and write them out precisely (avoiding such concepts as “of order aleph-null” and abuse of the hyperreals). Maybe even post it as an OP.>>

To this I responded:

KF, 112:>>Permit me to amplify, that first the oscillating universe models have fallen to entropy rise challenges.

Further, the observational data on the only actually observed cosmos points away from re-collapse to expansion, and as was discussed earlier with you, is fine tuned, on some calcs to 2 parts in 10^24 on density at 1 ns post singularity, with hints of yet finer tuning at earlier points.

Beyond, Spitzer summarised arguments that the transfinite cannot be traversed in separate finite steps. He did not merely assume.

Above, the exchange we have had is about precisely that.

You have admitted that you are unable to show such a traverse, and are now adverting to oscillating models that have failed.

I have taken time step by step to put the challenge in terms of completing the arrival at the present; in the face of many objections on your part.

You have spoken of how at any specific point, already an infinite number of steps is complete. I have set about constructing a distinct whole number A at transfinite distance in steps from an origin, by (in the end) using *some m –> 0, an infinitesimal such that 1/m = A, a transfinite whole number where A = W.F is such that F = 0, the fractional part vanishes*. The focal task then becomes traversing onward from A to 0, envisioned for the moment as the singularity, from which onward we go to now, n. Where as you objected to negatives [though how that was used was explained] I use asterisks to show the finite up-count since the singularity. Of course the lead ellipsis indicates that *A is not the beginning of the steps we may identify and list as a succession, it is preceded by an arbitrarily and per your suggestion for argument even possibly transfinitely large and unending set of previous values*:

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

Such, of course was already outlined by way of making the way clear after successive objections.

The start point for a count is arbitrary, so* let us put the start at A and put it into correspondence with the naturals, i.e. this is in principle countable . . . as is implicit in stepwise succession as would happen with clock ticks*, one providing the basis for the next as energy is gated from a source and as positive, precisely lagged feedback is applied:

A, (A less1), . . .

0, 1, . . .

*Given that the traverse from A to 0 is transfinite, the task here is comparable to counting up from 0 to a transfinite in finite successive steps*, which is a supertask that is unattainable. (And I have taken the step of identifying A as a specific number a reciprocal of a number close to 0 [as the hyper reals approach takes to identify what an infinitesimal is, only in reverse], to avoid all sorts of issues on what does subtraction mean with a transfinite. Such will of course be of at least the order — scale if you will — of aleph-null from the origin at 0. I take it that we can accept the reasonableness of infinitesimals close to but not quite attaining to zero; such being foundational to a way to understand the Calculus.)

For, *once we count 0, 1, . . . n, we may always go on to n +1, etc in further steps, always being finite*.

The evidence is that *traversing an infinite succession of finite discrete steps is a unattainable supertask*, precisely as Spitzer sums up.

The worldviews significance is this, that a contingent succession of beings, with each being b_i subject to on/off enabling causal factors it must have in place for it to begin or continue to exist, must be a part of a chain of successive and in context finite discrete causes. This can be in principle enumerated and compared to the step-wise succession, e.g. of clock ticks on a clock.

We then see that *the traversal of an infinite succession of such beings is to be doubted, on grounds of needing to arrive at the singularity then onward up to us*. From the singularity (for reference to current cosmology, actually any reasonable zero point would do equally) to us is explicable on a succession, but the problem is to arrive at 0.

This may then be multiplied by the challenge that non-being, the genuine nothing, can have no causal powers. There is not space, time-point, energy, mass, arrangement, mind etc “there.” So were there ever utter nothing, such would forever obtain.

We face then, the need for a necessary root of being to account for a world that now is.

Necessary, so connected to the framework of a world that no world can be absent such. As an instance, 2 must exist in a world W where distinct identity, say A, exists: W = {A|~A}.

A world now is, so something always was.

Following, frankly, the line in the classic work, Rom 1:19 ff (which I find to be enormously suggestive of a frame for a reasonable faith worldview), this world is a world in which we find ourselves as self aware, responsibly free and rational individuals; contingent beings subject to moral government and intuitively sensing the need to respond appropriately to evident truth about ourselves and our circumstances in a going concern world.

It is appropriate in such a context to ask, what sort of serious candidates — flying spaghetti monsters etc are patently contingent imaginary parodies that do not meet the criteria for necessary being and need not apply — can we see in making a worldview level choice?

After centuries of debate, there is one serious candidate, by utter contrast with non-serious parodies, and by contrast with the challenge of traversing the transfinite etc.

The bill to be filled looks extraordinarily like:

an inherently good creator God, a necessary and maximally great being worthy of loyalty and the responsible, reasonable service of doing the good in accord with our nature.

This is Candidate A.

Candidate B is: ___________ ?>>

It seems to me, that this is the underlying worldview level issue, and that as usual, the question pivots on just what is it that can be seen in comparing difficulties of start points, first plausibles. And of course as every tub must stand on its own bottom, DS is just as duty-bound to show why he thinks an infinite successive finite step traverse is impossible as he thinks Spitzer is to justify his assertion that such is not possible. And in the bargain, I think I have stipulated that m is infinitesimal and have taken reciprocal 1/m = A as a transfinite, specific whole number in reverse of the general approach used in defining hyper reals and using the concept that properties of reals extend in the argument to the hypers. **END**

Is an infinite past traversible in a causal succession of distinct, finite duration steps? Spitzer argues no, DS doubts his argument, I suggest Spitzer has a serious point.

PS: Observe how h functions here and especially the implications of h –> 0: https://diversity.umn.edu/multicultural/sites/diversity.umn.edu.multicultural/files/DifferentiationRules.pdf and compare the discussion per non standard analysis here: http://www.math.wisc.edu/~keisler/calc.html

KF,

As I mentioned in the other thread, you need to remove all references to “transfinite whole numbers” from your argument. They don’t exist, just as square circles don’t exist.

Edit: Ok, I need to qualify that: I’m taking “whole numbers” here to comprise the set {0, 1, 2, … }, as defined by the Wolfram site.

DS,

I suggest to you there is no more inherent reason why a number with no fractional part should not be transfinite than that one with such a part should not be.

After all, a whole number is effectively one where the fractional part is F = .00000 . . . And, given that there is still a matter of discrete causal succession at stake, the matter in question comes inherently in discrete states which can be assigned integer values.

For convenience, Wiki:

U/D: Keisler, p 159:

Seems reasonable to me. I would infer from that that I am dealing with a pair of reciprocals, A and m, such that 1/m = A, and where A is an integer of transfinite character.

If you can show different, kindly do so.

But the point remains, that you need to show us that we can descend to 0 step by finite discrete step from a transfinite. That is for sure what you have to mean by suggesting that at any given point there was already an infinite succession completed to that point.

It is that assumption of infinite succession you and others have made that needs justification, and I cannot see it having any for precisely the reason that we cannot ascend from 0 to the transfinite step by step either.

Whether climbing up or down the ladder at any point we are only ever at some n, a finite distance from the point we started.

Conventionally, we start at 0. But we obviously need not do so. For . . . -2, 1, 0, 1, 2 . . . –> do we not start elsewhere to add?

2 + 3 => 2 -> 3, 4, 5|

So, why not start at transfinite A = 1/m, m an infinitesimal close to 0, and descend?

If not, why not?

Just what contradiction of core characteristics is there, as happens with a square circle?

If so, how can one traverse the descending range to 0 step by step?

KF

KF,

This has nothing to do with fractional parts. The standard definition of the set of whole numbers is {0, 1, 2, …} (possibly without 0), none of which are transfinite. Each whole number has a decimal representation. If I ask you to name a transfinite whole number via decimal representation, you won’t be able to.

I have no issues with this. But then I’m not clear why you spoke of letting some number m (real, hyperreal, I don’t know) approach 0 to arrive at an “infinitesimal”, rather than simply choosing a single hyperreal infinitesimal directly.

As I’ve stated before, that doesn’t occur in my example. See below.

But we/the clock didn’t start, as my post #76 explains in the other thread. It is eternal, without beginning (but it could have an end without changing anything).

The ticks occur at these times in our coordinate system: {…, -2, -1, 0, 1, 2, …}

There is no tick A seconds before the present if A is a transfinite number.

Therefore there is no traversal A, A – 1, …, 2, 1, 0.

DS,

Kindly note Keisler, and the summary in Wiki that derives therefrom.

I believe it is fair summary that a real (or extensions thereof) can be represented with a place value, whole + fraction notation that is a shorthand for a power series on a base with each term b^n multiplied by a digit, d:

SUM [d*b^n]:

e.g. 19.78 = 1 * 10^1 + 9 * 10^0 + 7 * 10^-1 + 8*10^-2 + 0 *10^-3 . . .

In that context we use N = W.F, and of F is 0000 . . . then it is reasonable to speak of a whole. Or integer. Thence by extension hyperinteger.

All I am doing is taking that infinitesimals exist close to 0 and their reciprocals are hyper-reals. I take some m of that order and stipulate its reciprocal A has a whole part with F = 0000 . . .

Then, start counting down from A to 0.

I add i/l/o your comment, for emphasis, that A is not the first tick if you will, it is the one where the count picks up, only it is at a transfinite distance on the number line from 0.

Never mind your mathematical clock was ticking before that and per argument arbitrarily long before that, the issue now is to count from A down to 0.

Thus can be set in correspondence with 0. 1, 2, / . . and will run into the problem of stepwise traversal of a transfinite span.

KF

KF,

I’ll look at your post #6 more closely later, but I’ll say it again: my example does not involve counting down from A to 0, for some transfinite A.

Edit: I see you referred to that above.

It remains true that there is no clock tick a transfinite A seconds before present. If you think there is, identify its time coordinate in this set: {…, -2, -1, 0}.

DS, are you saying your clock has only been ticking for finite time? If not, it has to have been ticking a transfinite number of ticks past, and so would have been ticking at A if such were possible. KF

PS: I believe there is an answer already given, in the context of hyperintegers if you will, cf 4 on such — whole, transfinite numbers. Note the OP:

. . . A, (A less 1), . . . 2, 1, 0, 1*, 2* . . . n, n now.

A = 1/m. m an infinitesimal whose reciprocal will have fractional part F = 0000 . . .

F/N: The significance of this issue is, if a space-time plenum is all there has been, there must have been a past temporal causal succession. Where, either it is finite and something has come causally from non being at a start point or else a transfinite span of causes has been traversed in succession. Neither of these is attractive as a worldview option, indeed there are questions of incoherence or absurdity. The pivot is, why is there something rather than nothing? KF

KF,

It has not been ticking for only a finite time. The time coordinates (in seconds) at which the clock has ticked before present comprise the set {-1, -2, -3, … } (the negative integers). The set is infinite, but none of its members is transfinite.

KF,

I think the hyperreals will end up being not relevant but in this expression:

From what set are the b, n, and d drawn from?

DS, sorry but a set of integers that spans to infinity will have members that are transfinite. We cannot count to such members or count down from such (what the ellipsis in part indicates), or even write them down but we can point to and construct such on reasonable terms. That is how I come to highlight what I symbolised as A. KF

PS: Do you recognise place value notation and the power series it represents? Such is a commonplace of computing and digital electronics as well as mathematics. I used b for base as stated. d is the corresponding set of digits, commonly decimal, duodecimal, binary, hexadecimal and sexagesimal. n is the power of the base corresponding to places, in primary school, . . . hundreds, tens, units, tenths etc, in higher levels relevant powers of the base.

KF,

I have to believe we are not understanding each other here. The set of natural numbers (say) has cardinality aleph-null but has no non-finite elements. Do you disagree with this? [Edit: I’m interpreting “spans to infinity” in the obvious manner, I believe, although I wouldn’t use that phrase].

Of course. I’m asking you to tell me what kind of numbers these are. Rationals, integers, something else?

I am sorry, but I am trapped in the infinite now and I hear only one clock tick.

DS, if a set that effectively counts (i.e. successively lists) from 0 or 1 in succession has in it transfinitely many members in aggregate, it must span to transfinite numbers, or equivalently span to infinity thus including members of transfinite scale. If ALL of its members so enumerated or indicated in succession by ellipsis are finite, the set cannot but be finite. Indeed, the ellipsis with an open end indicates, continued without any finite uppermost term, we can only point there. Beyond such, with all due respect some of your questions are now trivial to the point of being questionable, if we both agree that we understand the standard place value notation. Your earlier remarks left me in doubt in this regard. KF

KF,

What does it mean for a set to “span to infinity”? There have been so many misunderstandings in this discussion that I want to be sure we’re speaking the same language. Does the set of natural numbers (which are all real) “span to infinity”?

Do you therefore claim that any set which has only finite integer members is therefore finite?

This is absolutely false, if that’s what you’re saying.

My point in asking that question was the following: expressions of the form:

will always be real, if your b’s, n’s and d’s are real (and presumably b > 1). You will not obtain any nonreal hyperreal numbers in that manner.

DS, how many natural numbers are there, what is the biggest and last one and how do you distinguish listing in succession from a count. Last I checked N is subset of Z and onwards R thence C. KF

DS, to identify that in principle a real number can be expressed in a PVN form does not imply that such will be exhaustively possible any more than can counting count all naturals. We cannot fully write pi or e, but that does not mean they are not real. Nor does this answer to what the open ended ellipsis entails for listing N. And so forth. At this point, it looks like you are flailing rather than addressing your core challenge, to show how one gets to just now in effect count down from infinity past to now. I suggest this is an infeasible supertask and time cannot be infinite in the past. I add, nor can a chain of successive causal events such as clock ticks or beings such as would chain back from the present of our world to the singularity and to whatever is beyond such. And yes this cries out for necessary being of eternity to be acknowledged as real and the root in which time comes to be. KF

PS: You have not engaged the definitions of a hyperinteger given this morning in 4 above. Those are crucial.

I’m probably way in over my head, but is counting down from infinity past to now the same thing as counting from now back to infinity past? I see the problem with the former, but not so readily with the latter.

Edit: To a certain extent, framing the issue as counting down from infinity past to now would seem to assume the very same starting point (named as infinity past) that it is trying to prove. To my quite possibly under-educated mind in any case.

Further Edit: Put another way, the idea of using infinity past as a starting point seems like a non-starter (if you will). If you can’t reach it from here, then you can’t start there. Similarly, you couldn’t use infinity future as a starting point. The starting point can only be either the present or a finite number of steps from the present.

Final Edit (I promise): It seems like the argument is being framed as:

Assume a beginning a finite number of seconds ago

vs

Assume a beginning an infinite number of seconds ago

But I think what is being argued is:

Assume a beginning a finite number of seconds ago

vs

Assume no beginning

KF,

There are infinitely many natural numbers. The set has cardinality aleph-null. There is no biggest or last one.

Do you agree? I’m asking for a direct yes or no answer here, because I sincerely can’t tell, based on your above posts.

To me,

means that any set consisting of (say) finite integer values must be finite. Is that not what you meant?

When I “count” a set S, I am usually referring to setting up a correspondence between some subset of the natural numbers and S. Listing in succession does not imply knowledge of a correspondence.

And yes, I agree with the chain of set inclusions that you described.

Surely you are aware of the fact that the reals are in one-to-one correspondence with non-terminating decimal expressions (and a + or – sign)? No non-real hyperreals can be expressed in that form.

They aren’t crucial, because I stipulated that the clock ticks occur only at (real)

integertime coordinates. No hyperreals involved here. If you produce an infinite hyperreal A, then a tick did not occur at time -A seconds.DS, please. You know better, far better. If there are infinitely many naturals, there will be no last member, hence the ellipsis. If you look I built that in to the down count challenge, you are setting a start count from A which is itself not the actual first, but is a convenient start for a count. Then the problem is it is transfinitely remote from 0 and a count process will always only reach a finite number. In short the transfinite range cannot be bridged to get to the big bang singularity on the assumption time has an infinite past. And you obviously have not reckoned with why A is a transfinite integer. That allows us to set a start point for the down count at a transfinitely remote INTEGER, and contrary to your it’s like a square circle stunt, they exist. Likewise a set 1, 2 has span 2, 1, 2 . . . 10^500 has 10^500, The full set is infinite and so must embrace some transfinite A etc. KF

Phineas, whether you try to count down from a transfinitely remote point or up to it makes little difference. A transfinite span cannot be bridged in discrete finite steps. For going up, count 0, 1, 2, . . . n. At any n arrived at the result is inherently finite and can go on to the next finite n + 1 and so on. You will never reach the transfinite stepwise. A is not a beginning, cf the OP, it is just a convenient but transfinitely remote point along the way to start counting to show the problem of bridging to 0 from an infinitely remote causal chain and temporal past. KF

PS: I clip the OP:

kf:

I think I understand. My initial thought was that the notion of a “bridge” may be misguided, because it assumes a far end point that doesn’t actually exist. Any bridge to the infinite is a bridge to nowhere. Or everywhere. Maybe that’s the point, but it seems to me that you’d need to assume an infinite bridge to reach an infinite point. To get to no beginning, you’d need a bridge with no end.

On the other hand, a bridge with no end may not be the same as an infinite number of bridges, which is what the discrete time steps require. With an infinite number of bridges, each one has to have two end points, so you can never have your bridge with no end. And without a bridge with no end, you’ve no way to reach no beginning.

I know I have eternity in my heart, but it can be difficult to get it into my head.

KF,

First, I am trying to be as forthcoming as possible and answer any questions you pose, because we have had a difficult time communicating. When you decline to answer my direct yes/no questions, it make it hard for me to understand your position.

I take it you are affirming that there are infinitely many natural numbers?

Do you also agree that all natural numbers are finite? (yes or no, please).

Is A a real number? If so, what is its value?

Is A a nonreal hyperreal number? That number is not on my time axis, hence I don’t have to account for anything. The clock ticks occur only at real integer time coordinates.

No transfinitely remote real integers exist. Do you agree? Yes/No?

Again, nonreal numbers are not on my time axis.

Ok, even if you decline to answer any other questions in this thread, please answer this. Does the set of real integers, commonly denoted Z, have any infinite members? Yes/No?

DS, you are the only person who has claimed all natural numbers are finite. You need to show cause for that claim. I think there is a serious problem with that as the number of naturals is the same as their in principle count that is you are looking at transfinite cardinality pointing to transfinite members as the open endedness 1, 2, 3 , , , points to. I will not go over A again enough has been said, it is transfinite, with fractional part 0000 . . . and is thus an integer, i.e. 1/m will here give a whole number. We are still miles from your showing cause that a count down to zero from transfinite order of magnitude is feasible, you have simply asserted as though it is. That is dubious. KF

Phineas, the attempted bridge or rather cascade of bridges cannot be built. KF

KF,

Thanks for answering. Yes, I do claim that all natural numbers are finite. It’s likely this the root cause of our disagreement.

DS, the problem I see with that set and the claim that all its members are finite yet its cardinality is transfinite is its very definition is linked to the counting numbers, so if it is NOT transfinitely large only then will it not have transfinite numbers. And, no I have serious doubts about the trick that oh a given number n will always have a yet higher one. That is saying transfinites do not have yet higher order transfinites. Flip that into the infinitesimals and see what that does to calculus. KF

KF,

Have you looked at the rules for cardinal arithmetic?

How many non-ticks between each tick, that’s what I want to know.

Ds, let us get back to centre: how do you descend from an infinite past in steps to the present? Why should anyone accept that such is any more feasible than counting up to the transfinite? In effect saying that at any time you already descended sounds a lot like begging the question, and in the face of a serious issue. And no given that one can go m as an infinitesimal near zero, 1/m = A, a transfinite, I see no reason to imagine that one cannot reasonably identify specific transfinites and even distinguish their scale. Notice this is not obtained by counting. I simply constrained m such that the operation yields a hyper integer. KF

PS: Obviously this set of issues is not for a popular discussion.

I dunno about that. My mother’s father repaired clocks, but I can’t recall one that didn’t require winding.

KF,

I don’t believe you can count down from aleph-null, the smallest infinite cardinal, to 0 in discrete steps, if that is what you’re asking about. Aleph-null is a limit cardinal, which renders such a thing impossible.

I would suggest you do some more research on whether infinite natural numbers exist. Surely such a fundamental question has been answered already?

One suggestion: Look at the well-ordering principle for the natural numbers. Then consider if that’s consistent with the existence of transfinite natural numbers, from which you can’t count down to 0 in finitely many steps.

I’ve been taught that an actual infinity cannot exist in this universe. Also was taught that if something exists now then something has always existed because out of nothing nothing comes.

An interesting story about that. I posed the question true or false about the existing statement to some college students. I couldn’t believe that some of them actually thought the statement was false. They were atheists. I had to explain to them that something had to have always existed, but it couldn’t in this universe. The multiverse of course came up as it almost always does and I had to explain to them about that there is no proof of that and that any way an infinity would have to be crossed to get to our universe. I just explained that there had to be an Agent Who would exist in a timeless existence.

I find the subject to be beyond our capability to visualize it or comprehend it. We can conceive of it, but cannot explain it fully because we are creatures of time and space. The kids left as agnostic instead of atheists, a good step. I did explain to them about how the God described in the Bible fits and they did show some interest. I hope it bears fruit.

DS, we are still at the central issue. The problem is that in effect a transfinite succession of discrete finite steps would have to be bridged to get here from an infinite past. It seems there is now a belief this is possible, likely driven by the prevalence of evolutionary materialism, which must either draw the origin of our observed cosmos out of non-being or else find some form of infinite succession in a causal chain to the present. Neither is attractive but the discussion of the transfinite is patently filled with difficulties. KF

PS: At this point, I will say, set aside definitions of what is or is not a natural number and the claim that as it has no end and each named has a successor each is somehow limited and finite though the set as a whole is of transfinite cardinality. I think that is already a warning flag that something is wrong. Instead, take some small number p and do the reciprocal operation, which will be a large number, P. For most cases P will not be a whole number nor can it be fully expressed in place value form, but in certain cases it will be. Now, let p go to m, an infinitesimal that when the reciprocal is taken gives rise to a number A which is also such that it has no fractional part. That is, it is a hyper integer. The descent in steps from A to zero will face the same challenge as counting up from zero. Namely, at each successive state we will only have completed a finite set of steps and cannot go to the next and find it transfinite. That is we cannot complete the walk of counting in either direction. Linked, we may define the counting numbers — I am being strictly descriptive — 0, 1, 2 . . . and point to their succession but cannot complete it. That last part is crucial. I remain of the persuasion that as each successive subset produced by counting will have as final member a value equal to the cardinality so far [and we evaluate equality and cardinality by putting members of a set of interest in one to one correspondence with the successive counting numbers until exhaustion or else reason to believe it continues forever], the set of counting numbers as a whole will have a transfinite cardinality, pointing to an endless membership.

JM (attn DS),

it seems we have been inundated as a civilisation by a worldview that either must derive everything from nothing or else must posit an effective, infinite finite-step causal succession. Neither of these claims is attractive — and that for cause — but the matters are deeply clouded as we have seen above and previously.

I think most can see that counting (or any similar stepwise finite stage succession, e.g generations, seconds of time ticked off by a clock etc) up to what is beyond the finite cannot be completed, but because of claims that such has been completed to arrive here cannot recognise that front ways or back ways the same span would have to be bridged.

Right now, I am looking at the implied view that such a succession would have to have been bridged without resort to any non-finite value at any point and success can in effect be declared and taken as start-point; and of course if you question, you are suspect.

I further find that the usual terms are now too loaded with issues to be useful in discussions . . . never a good sign.

That is why above and previously, in trying to reason the matter out I reverted to taking 1/m, m an infinitesimal, and generating a number of transfinite value that has no fractional part. Descent from the resulting A in finite successive steps would seem to face the supertask challenge.

But that such an A can be constructed is patently fraught with issues.

Not least we find a claim the naturals are all finite but the set constructed from them in succession is of transfinite cardinality without any member of that character. That, I find troubling and

Such would lead to the point that we can only actually construct or count to or fully represent some finite numbers (and going beyond counting numbers the number line viewed as continuum will contain many members we cannot wholly represent, such as pi etc — but obviously can exceed with a countable number, even as 4 is greater than pi . . . ), but we can point beyond in accord with a trend, to the transfinite.

Hence, the utility of open-ended ellipsis.

Likewise, when we see that a counting number n is reached by counting: 0, 1, 2, . . . n, equivalent to successive addition: 0 + 1 + 1 + 1 . . . + 1 with n 1’s, we see that closing the ellipsis in fact step by step implies finitude. Which comes out with the completion. And/or with exceeding it, at n + 1. Where also this has nothing at all to say about whether all numbers of like whole character can be counted to. Howbeit as one may always succeed

a given number attained by count[note the constraint, you have to get there . . . ], we see that there will be finites that can be produced in succession; we also see that we cannot exhaust the set of counting numbers as a whole, it is transfinite in scale. It is the next step that is dubious in my view as at now, to then assign that all counting numbers are therefore finite, as that begs the question of actually counting up to them then past them. All counted numbers or exceeded numbers are finite, but to what I can see to date, that does not then extend to the claim that all numbers that are whole — fractional part uniformly zero — will be finite. Indeed, as the reciprocal of a fractional number may be a whole number, it seems reasonable that some p of that character may be such that 1/p = P a whole number. Now, allow p to go to an infinitesimal, an exceedingly small number m while retaining the same sort of relationship that its inverse 1/m = A will be a whole number, what has been called a hyperinteger, one that is transfinite. I find this reasonable, as A will be a whole number that is beyond counting. It belongs to the set of whole numbers but cannot be reached or exceeded by directly counting. I would be prepared to accept that the set of whole numbers — regarded as “natural” or as “[positive] integers [with zero]” — can/should be reasonably viewed as enfolding:{0, 1, 2 . . . A [= 1/m] . . . }

But obviously, that point is a personal view question and/or perspective and is open ended.

Perhaps, this is why the hyper reals are often portrayed as beyond all reals and their reciprocals the infinitesimals as below all reals? That gets around this debate by definition; but at the price in my view of being somewhat artful with naming conventions and framing of terms. I remain open to suggestions.

So, I think the best — least prone to confusion — answer to the matter is that

You are also right to point to the absence of observational warrant for multiverse claims.

KF

DS,

I will note to you that I have decided to add a category, stirring the pot for exploratory thoughts.

I put it forth, notwithstanding, that:

KF

KF,

Regarding #38, if done in a finite amount of time, I guess either of those would be supertasks (at least).

To address #36, I think we can’t set aside the issue of transfinite natural numbers and make any progress. The clock ticks are supposed to happen precisely at (negatives of) natural number time coordinates, so understanding that set is crucial. If transfinite natural numbers exist, then the clock example fails (from my point of view).

If I might recommend a quote from Cantor and two texts of reference:

“The actual infinte arises in three contexts: first when it is realized in the most complete form, in a fully independent other-worldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number, or order type. I wish to make a sharp contrast between the Absolute and what I call the Transfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite.”

(Georg Cantor, translated by Rudy Rucker, Infinity and the Mind, Princeton University Press, 2004.)

Note an excellent interdisciplinary study: Infinity – New Research Frontiers, Michael Heller, Hugh Woods, Cambridge University Press, 2011.

I participated in the original discussion, but only about the mathematics concerning infinity: I never paid attention to the argument by Spitzer that started the topic.

But now I see kf starts this post with this:

Is kf saying that Spitzer is saying there could not be an infinite past because if there was there would be no way we could be here now?

If so, since now could be any moment, that would imply that no moment whatsoever could exist, because you could never get there from “the start” of an infinite past.

And therefore, the fact that we are here now is a proof that time had a beginning, because if time didn’t have a beginning, no actual moment could ever exist (because you couldn’t get there). Or, more broadly, if time has an infinite past, then time can’t actually happen.

Is that Spitzer’s argument?

Aleta,

We should distinguish first an infinite series based on the procedures of calculus, which can terminate in a finite value due to the power of infinitesimals. L’Hospital and co.

Spitzer’s point is in effect about an infinite chain of finite causally linked steps and can be seen: http://magisgodwiki.org/index.php?title=Cosmology

esp: http://magisgodwiki.org/index......st_Time.22

the issue is to traverse an infinite span in finite steps.

KF

But have I correctly summarized Spitzer’s argument?

Aleta, the core is a couple of paras, as I just linked. KF

I have tried to summarize what I think his arguments are in simpler, more straighforward language. Have I correctly summarized his arguments?

Pointing me back to the original written material doesn’t help answer my question. I would appreciate your opinion, as the author of the OP as to whether I have correctly summarized those arguments.

1. Not only is it impossible to count back an infinite past amount of natural events, 2. To have an infinite amount of finite events means the materialist believes in the tautological oxymoron of “infinite finiteness”

Look folks, if it takes an infinite number of events to get to the present, the present could never be gotten to. Infinite regress logically cancels out any possibility of a “present” ever existing. It’s really that simple. If you don’t agree, you’re mentally impaired or playing games.

Mike1962 seems to be saying the same thing I wrote in 41, even more succinctly. Mike, I assume you also are summarizing your understanding of Spitzer’s argument – true?

F/N: Let’s do some clipping from Spitzer:

I hope that helps.

KF

PS: Note his use of steps implies finite discrete stages in cumulative succession.

My question to you, kf, is a simple one: is my statement at 41 a reasonably accurate summary of Pfitzer’s conclusion and arguments. I don’t want any furthur explanation – I want to know if my understanding is correct,

The OP started with the statement “that it is impossible to traverse an infinite past to arrive at the present.” I’ve read the arguments, and am looking to see if I understand.

So, kf, could you comment on my post at 41: does it say, in my words, which are simpler, approximately what Pfitzer is saying?

Aleta:I assume you also are summarizing your understanding of Spitzer’s argumentI didn’t actually read Spitzer’s argument.

But you do agree that your conclusion is that therefore an infinite past is impossible – true?

Yes. The idea is logically self-canceling.

Aleta:

I intend a response, but must note that Spitzer should be permitted to stand on his own merits, not whatever I (or you or a third party . . . ) may or do or did represent him as.

Spitzer’s central point is that the cumulative step by step process [adding from a start point and through a finite discrete step at a time applied in order, attaining to some particular total or else having attained to a result through such a step by step increment . . . ] is inherently finite as opposed to infinite.

Such a process is countable and as completed is counted [assuming some entity has kept it in track, such as a step counter or an odometer that tracks revolutions of a wheel etc], amounting to that finite total. Such a completed observable in principle past event or process is one that can be viewed as

“having occurred, or having been achieved, or having been actualized.”This, by contrast with the present or future.In that context, he points to an analytical distinction, that by contrast infinities

“within a continuous succession imply “unoccurrable,” “unachievable,” and “unactualizable,” for a continuous succession occurs one step at a time (that is, one step after another), and can therefore only be increased a finite amount.”In short, the transfinite cannot actually be completed in the past, or at present. Where by definition the future has not yet obtained, but we can be confident on the logic of inherent finiteness of a cumulative succession of finite discrete steps, that once a now future point is attained, there will not be completion of an infinite succession there, too. This is of course an assertion of reasonable faith — we look to the future based on the present and past with a confident and hopefully well founded expectation. Here, rooted in our confidence in the logic at work and in the correctness of the conceptions at work.

In that context, the claim to have completed such an infinite succession inherently must fail, or the distinction between the infinite and the finite is lost. Hence, the analytic contradiction Spitzer detects.

That is why he went on to state:

Asserting the completion of an infinite result by an inherently finite process is a contradiction in terms.

My own exchanges above and previously are in that essential context, though not directly due to Spitzer. You will recall my astonishment over the past few days to see that it is apparently commonly held that the set of counting numbers is full of inherently finite members and yet is of transfinite cardinality. I had always thought that it was appreciated that the set, being endless would contain the transfinite. That is why I took time to try to find a way not to run into various definitional issues and provide what seems to be a reasonable way to get a number with no fractional part but which will be transfinite, which for convenience I termed A = 1/m, m –> 0 but m != 0, m being infinitesimal.

To then propose a count from the indefinitely large and transfinite past, through A and onwards to 0 [say, at the singularity], thence now, n, will run into the problem that from A on to 0, a transfinite would have to be bridged by an inherently finite process. Where A is not a first point, it is a hyperinteger in the sense of being the reciprocal of an infinitesimal. I suggest it is of the order of magnitude Aleph null, but that is a separate matter.

Back to Spitzer, I think he knows a considerable body of thought on the subject and has sought to avoid tangles by going to the analytical point directly.

Now, we can look at your summary — this sort of conceptual thicket is anything but a simplistic yes/no circumstance:

I think, partly correct, partly not on Spitzer’s focal point. Yes, he does not think we can have achieved the present through completing a transfinite traverse step by step, which is an inherently finite process. No, he is not saying so much we would not arrive at the present as that to claim completion of a transfinite result by an inherently finite process of step by step accumulation is an inescapable contradiction and thus absurdity. Thus, his remarks on failure of imagination.

Now that I am more aware of the conceptual thickets at work, I think there is significant wisdom in his stance.

By its very meaning, the infinite cannot be attained to by a process that is inherently finite, even if cumulatively so.

KF

PS: secondary captcha and white screened.

kf writes, “I think, partly correct, partly not on Spitzer’s focal point”

Good.

kf writes,

Again good, and in agreement, I think, with what I said.

kf writes,

I don’t completely understand this sentence. Is there a difference between “he does not think we can have achieved the present”, which you said yes to, and “we would not arrive at the present”, which you said no to.

In both quotes you state essentially the same argument, which I understand – that we can’t complete a transfinite traverse step by step in a finite number of steps, so to think we could have arrived at the present is an absurd contradiction.

So I don’t understand what you are saying “no” to in the second quote that is different than what you said “yes” to in the first quote.

F/N (attn DS & Aleta et al): I am thinking a bit on what approach we can take to further characterise an A as I have put up above.

So, in exploration, let us now look at ordinals.

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:

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.

So, we see a supertask, and the contradiction Spitzer speaks of seeming to appear: completing a transfinite process stepwise by finite steps that at any point can only amount to a finite value. Indeed, such is more or less the point being argued in claims that all natural numbers are finite even in the face of a set of transfinite cardinality.

There is no problem in the count from 0 to n*, but there is an issue with A to 0 and also by extension down to A also.

Of course some will insist we are here and so we can postulate that a transfinite down count is complete, so counting processes can become transfinite and complete.

That needs to be noted as an assumption not a fact, and one open to challenge.

Serious challenge: how does one show a count process spanning the transfinite?

Again, open for discussion.

KF

Aleta, his focus is not on we cannot or do not reach the present — which IIRC is one of Craig’s arguments in the context of the Kalam cosmological discussion — but on there being an analytic contradiction between positing an inherently finite process and its claimed achieved traversal of a transfinite span. And I am old fashioned enough to prefer strongly transfinite to infinite. Note how I have just put up a F/N that explores such from my perspective. KF

In response to 56, kf, at least for me, all that continued explanation is unnecessary. I understand the argument.

Aleta, thanks. I am continuing on an earlier line of discussion in which I sought to identify something useful as a hook point for down count that is credibly an acceptable number transfinitely remote from 0. Namely A which could be seen as 1/m, m sufficiently close to but not 0 and also set up such that its reciprocal is whole, that A would be transfinite, some sort of hyper integer. KF

kf writes,

I think a word was missing, so I inserted what I think you meant.

That is, if I understand you correctly, his focus is not so much on the conclusion – that we could not reach the present, as it is on the argument as to

whywe could not reach the present.I think I understand everything I need to understand now.

Aleta, yes. He is saying there is an analytical clash between a finite process and a transfinite span. KF

KF,

Addressing your posts above,

1) Indeed it is commonly held that the natural numbers, which comprise an infinite set, has only finite members.

You can prove this quite quickly. Assume there are natural numbers n such you cannot count down from n to zero in finitely many steps. By the well-ordering principle, this set has a least element, call it k. Note that k is not zero, so k has a predecessor, k – 1. But you

cancount down from k – 1 to zero in finitely many steps, hence you can count down from k to zero also in finitely many steps (it just takes one more). This is a contradiction, so there are no such infinite natural numbers.2) Letting a real number m approach zero never gets you an infinitesimal. There are no real-number infinitesimals. You just get m’s closer and closer to zero, and the resulting A’s will be larger and larger, but still finite.

3) Whether you want to use ordinals or cardinals, the clock I described does not tick at any of these infinite time coordinates, so I don’t have to worry about bridging any gap between little-omega and 0. This discussion of the infinite is interesting in its own right, of course.

DS, I must disagree. It seems to me that if k is indeed transfinite counting down from k is a finite process and will not span to 0 despite the ordering succession. Please see 56 above on this relative to A, where I align A, A~1 etc with 0, 1#, 2# etc and as that process will be finite it will not span the relevant part. As g in A = w + g can be as big but finite as you please it can take in any k – r so long as r is finite, leaving w untouched, where w is of cardinality aleph_null. In your comment just now that issue does not seem to have been resolved so it looks to me like a begging of the question. I do understand that a count up from 0 will terminate at a finite and bound point n such that we then go n + 1 to the next. But that is part of my problem, running beyond is still finite, the transfinite is not spanned. As to the clock does not tick at A, A is in the cardinality zone of the in principle countable transfinite aleph_null. KF

Aleta, 62 vs 63 is part of the context of 56. KF

I haven’t really understood DaveS’s point for quite a while, so I’ve not been thinking about that.

Aleta, it seems that DS reflects a common view that sees an infinite past sequence completed now or finitely remote from now or at any given point. He also reflects the view that at once all natural numbers are finite and the set of the naturals as ordered {0, 1, 2 . . . } has transfinite cardinality. For which 62 was intended as a warrant in brief. My concern is how to symbolically, algebraically and logically express my concerns with such a view. Especially in a context where the very terms in use are now freighted with meanings shaped by the perspective DS reflects. KF

KF,

There are no transfinite time coordinates in the clock example, so there is no counting down to 0 from a transfinite number.

Do you still maintain that there exist transfinite natural numbers? If so, what is the flaw in my proof?

The fact that all natural numbers are finite is really not in question. This has been resolved long ago.

See here for another proof.

KF,

Just noticed this:

I haven’t seen a proper construction of A yet. You refer to a number m approaching 0, and for each such m, A = 1/m. These A’s don’t converge to any number, finite or otherwise.

Certainly you have not displayed anything in the “cardinality zone” of aleph-null, whatever that would mean. A cardinal number either equals aleph-null or doesn’t. If your A is less than aleph-null, then it’s a finite cardinal number. If it’s greater than aleph-null, then it’s greater than the cardinality of the integers, which would be absurd, if I understand your argument.

DS, kindly look above. If your clock has been ticking from the infinite past is has been ticking for a transfinite number of ticks and is subject to the concerns above. You are trying to traverse the infinite in steps, and to do so are suggesting that it is infinite now but oh no it isn’t. This sounds very much like the incoherence Spitzer warned about. KF

PS: m is infinitesimal and constrained to be multiplicative inverse of A so A is in effect a hyperintegral, which in turn is A = w + g, which would be of cardinality aleph null. As was already indicated. The problem with your link is that it in effect implies that the count up to n and past n can be extended indefinitely and will traverse the full set while being feasible step by step. That is how it poses the claim every member is finite and the cardinality of the whole is transfinite. There seems to be a problem here as the transfinite count is exactly what the challenge is to complete. And a number of form w + g will be transfinite.

KF,

I agree that it has been ticking for a transfinite number of ticks.

Absolutely not.

I suggest you think more on the issue of the existence of transfinite natural numbers. Until you are able to see why they cannot exist, we won’t be able to come to terms on this issue.

There are no real number infinitesimals.

Are you referring to the proof I linked to? The one authored by Terence Tao? I’m sure you’re aware he’s a pretty bright guy. He’s not in the habit of making simple mistakes.

If you can find a

single proofof the existence of infinite natural numbers, or even a serious discussion of it, please link to it.In the meantime, here’s a question: Form the subset of all

finitenatural numbers. What is the cardinality of this set?KF,

Another way to explain the idea behind my proof in #62 is this: Let A be the least among infinite natural numbers. Then A is the successor of a finite number A – 1; the sequence A – 1, A – 2, …, 0 has a finite number of steps.

Hence in your view, one

cantraverse the infinite in finite steps, namely: A, A – 1, A – 2, …, 0. Just tack A onto the finite sequence above. Which is very ironic.DS, nope again. You will not get up to to A by a finite stepwise count process — and counts will be finite as actually carried out — or down from it by same. That’s why you see multiplicative inverse of an infinitesimal m. Just the additional property, w + g, so in an ordinal scale of cardinality aleph null. At 56 it was shown how count down will not access 0, indeed as A = w + g also, g can be finite but large enough to accommodate any finite step back process as at 63, leaving w on the table. Recall w is omega the first transfinite ordinal, of cardinality aleph null. KF

I have read the comments here and all I have seen is KF and DS measuring dicks. I don’t even have a dick but I can tell when both of you are arguing about and abstract, and valuable, mathematical concept as if it really meant any thing with respect to the OP.

Get over yourselves.

Ginger, the relevant past would be a chain of causes as steps. The possibility of a completion of an infinite chain of steps is in fact a key issue to the claim of an infinite past completed in the present. Unfortunately, such is closely connected to the mathematics of infinity, which makes that side of the issue hard to follow. Mathematics, being understood as the logical study of structure and quantity. My concern is, it looks uncommonly like a question is being begged. KF

KF,

Well, I was laying out a reductio argument, to highlight the difficulties with your position.

Ponder this: You are forced to accept that there is an infinite natural number which is a finite number plus 1.

Does that make any sense?

The flaw with Spitzer’s argument

Summary: there cannot be an infinite past, because if there were, no finite number of steps (seconds, causal connections, or whatever) could ever get to the present (or any existing moment). Therefore, time must have a beginning.

The problem is that the number line is being considered an accurate model for time, which is, in my opinion, unwarranted.

Let me explain. Mathematics by itself, is a pure abstract system. To apply mathematics to the world, one must first create a model which maps elements of the math to elements of the world. Then one tests conclusions and predictions made by the math with the real world: if those results agree, that is evidence that you have an accurate model. If they don’t, you revise or refine the model (after making sure you’ve made no mathematical mistakes.)

Now, one of the most useful mathematical inventions ever made was the Cartesian plane, created by Descartes in the 1600’s. The basis of the Cartesian plane is two infinite perpendicular number lines, the axes, meeting at the common zero point of the lines, the origin. This was later expanded to three dimensions by added the third mutually perpendicular axis.

In addition, time came to be considered, likewise, as an infinite number line, both as a parameter in Cartesian space creating motion, such as a point moving along a curve, and later as a fourth perpendicular axis, albeit imaginary.

This framework proved very useful as both a model of motion in space, and of other types of changes in time: it was the default framework for both calculus and Newtonian physics.

A key idea of the framework is that space and time are there irrespective of anything in them, or of any movement. Even if we removed all content from our universe and anything beyond or universe, this universal framework of space and time would exist.

We now know that this is not true. Space, time, and motion are inextricably bound together in non-Cartesian ways within our universe.

Furthermore, assuming that our universe sits in a Cartesion framework – that those nice tidy perpendicular axes extend infinitely beyond the universe, is entirely unwarranted. If space and time are as different from the Cartesian model as they are in our universe, how can we begin to know what they are, or even whether they exist, in whatever might be beyond our universe, (if anything).

So the mathematical discussion about infinity that’s been going on here, while interesting, can’t really tell us anything about whether time started or not, or whether that is even a meaningful question. We have no reason to believe that modeling time as an infinite number line, back beyond the start of the universe, is an accurate model, and no way of testing whether it is or not.

So Spitzer’s arguments, no matter how accurate they are mathematically, do not lead to any reliable conclusion about the true nature of time.

Aleta,

pardon but the issue is not so much a model of time as of a succession of causal entities and processes step by step.

Yes these occur in time, but in causal succession.

For simple example, consider big bang –> formation of relevant matter and of stars and galaxies thence planetary systems, including our own, onwards down to us being here. The issue then is is that spatio- temporal causal succession an infinite regress that completes in the present, or is this inherently problematic.

Spitzer’s point is that a step by step succession that is completed or completed to date at any rate is inherently going to be finite. And by contrast an infinite pattern of succession is beyond the finite, is inherently beyond completion by traversing the transfinite step by step. Such would include the succession I have outlined at high level just now, and whatever lies beyond it on an infinite past proposal.

In a previous thread I already pointed to the evidence of a big bang as a start point of reference at finite remove, and to the issue of accumulation of entropy multiplied by evidence of fine tuned density of space and the implicaiton of continued expansion not recollapse and proposed oscillation.

Further to this, oscillatory models will run into accumulation of entropy.

Consequently Spitzer’s LOGICAL argument is very relevant to the issue, and it implies an associated set of issues on the mathematics of proposed infinite stepwise succession.

My discussion of A is in the context of showing the core problem with such a succession, and how the traversing of the transfinite becomes an issue.

KF

DS, It looks to me that you have inadvertently begged the question at stake, traversing the transfinite in stepwise finite stages. KF

KF,

Can you tell me where this happened?

Here are the steps:

Where’s the error?

UD Editors: Ginger Grant is no longer with us.

DS,

The basic problem you still have to address is to bridge to the infinite one finite step at a time in cumulative succession.

If you add one to a finite value it is still finite, you have not bridged to infinity.

Let me put it this way, take some finite value f, and add to it any stepwise achievable neighbourhood s. The resulting value f + s will still be finite.

Likewise take a value of order w + g, and step down any finitely achievable range s [where g, though finite, can for convenience be such that it is always in excess of s however large s may be . . . allowing us to keep the subtraction as between finites and avoiding issues tied to “subtracting” a finite from a transfinite], we still will be within the range of cardinality w, which is aleph null.

The transfinitely large is a zone we can postulate and discuss as to consequences for mathematics and logic but we cannot attain to such by any finite process of cumulative steps, or, once there cannot depart from it by such a process.

Yes, there is an ordered succession of counting numbers — and for the real world of chained cause effect bonds, a succession of events and results that may be tagged as to number in sequence — but the very point of proposing a transfinite result is that it is beyond what can be attained by the finite process. And once in concept we are there, I would argue with many others that while we may consider what happens there, we cannot bridge back to the finite by a finite stepwise process.

You will notice, the process I used to bridge, taking the multiplicative inverse of an infinitesimal. A small number has a large reciprocal, and an exceedingly small number closely approaching zero will have a correspondingly large reciprocal. Given that non standard analysis exists, we can treat infinitesimals (which I gather Cantor did not like at all) as credible. Given the all but zero nature of such, their multiplicative inverses will be transfinite. So, I have proposed a conveniently sized infinitesimal m that on multiplicative inversion gives a whole number A such that A is also in the cardinality zone aleph null. This then gives us 1/m = A and also A = w + g, g some finite value.

Notice how one bridges onwards to higher cardinalities, taking power sets and evaluating their cardinalities. Again, one envisions but cannot complete the stepwise process of ordinals to go up in scale. The way to get there is to do a far more powerful operation as the deduced cardinality of the set of all subsets of a set C of scale c from {} to C itself is 2^c. Thus from aleph_null, aleph_one, aleph_two etc. And that etc implies again a stepwise process that cannot be completed in finite cumulative steps, even steps as powerful as these

~~seven~~transfinite league boots.If you step down from a transfinite value of order w + g with a finite number of steps, you are still in the zone of cardinality Aleph null.

Now of course, all of this is as I can see at this stage of thought, I am fully open to change were something there for me to see a finite step succession based bridge to the transfinite.

In short, I am seeing the infinite as a mental-world concept that we may contemplate. On such contemplation, it is quantitative and structural relative to numbers and things tied to numbers, but is physically and even logically/conceptually unrealisable in stepwise finite step cumulative processes.

Perhaps, that is the problem, there is a different conception of the infinite at work.

Perhaps.

It seems, however, that — consistently — the problem highlighted by Spitzer obtains:

For the now, here I stand, on the force of what I can see. Perhaps, I see faultily, but to correct that, I ask for good reason grounded on a basis that answers the concerns I have made.

KF

KF,

I agree with this last statement. Adding one to a finite number should always give you a finite number.

Now take a look at my post #80.

I have shown that the existence of infinite natural numbers is inconsistent with this statement.

My conclusion: Infinite natural numbers do not exist.

Do you therefore agree?

If you’re not going to answer these very direct questions I’m posing, then there’s not much point in having this discussion.

DS, no you have not, you have unfortunately failed to bridge to the transfinite. And the problem is that the definitions you are using are caught up in that issue. Demanding yes/no simplistic answers in a context where such conceptual challenges are at work is likely to be useless or worse than useless. The pivotal point is that it seems very much the case that whole, ordered, counting numbers can reasonably be extended to the transfinite but cannot be accessed by inherently finite step by step processes from 0. Likewise once at that level, descent to the finite thence 0 is not feasible on such step by step processes. Unless this premise is shown to be false the problem cannot be addressed as you have sought to. Spitzer’s point is serious. KF

KF,

Then kindly point out the flaw in my proof.

Edit:

Specifically, what is wrong with post #80?

DS, The problems have been repeatedly pointed out. The core issue is the nature of the finite vs the infinite or transfinite, and the use of inherently finite step by step processes in attempts to bridge from 0 to the transfinite or from the transfinite to 0. Providing definitions of natural or real numbers influenced by this problem will not help. So far I am seeing that the transfinite makes sense as a mental, mathematical construct that allows us to address foundational issues, but the bridge from the finite to the transfinite and the reverse is not accessible to a step by step finite stage counting-driven process which at any finite stage n followed by further steps to s which is within a finite neighbourhood of n so will also be also finite, will in principle fail to access the transfinite. The picture of a lowest transfinite sitting next to a finite accessible to zero, one step away, becomes questionable. KF

KF,

If there are genuine problems, they should be manifested in my proof in #80, which you stated does not succeed.

What are problems

in post #80 specifically?Edit:

Yes! It’s completely ridiculous. That’s why your position is untenable.

DS, the problem is, that is exactly what is NOT being pointed to, the portrayal of a transfinite next to a finite one step away fails, and it is also not what is being stated by Spitzer. The very point of the transfinite is just how far beyond reach of a finite process it is. KF

daveS, just give up. After many years of lurking here, I have yet to find an instance where KF has let other people’s logical arguments influence his own beliefs. He is right and you are wrong. And if math agrees with you, then math is also wrong.

hrun0815,

That’s good advice. At this point, I am going to take it.

You’ve put into words exactly what I was thinking:

And now wait the long word salad that confirms exactly what I said in #89 …

HR, you have moved from the topic to personalities. KF

DS, I think the point is that there is a key issue on what is the transfinite. The reason why we speak of the set of counting numbers as that is, they are endless, literally. Aleph null is a metric of that endlessness, not of some final value. That is the context in which no finite process can attain to them, and the stepwise count or causal chain process is precisely that, inherently a finite process. Until that is duly weighed, there will be no acceptance of the distinction and limitation. The only way to actually reach such levels is by a process that is more powerful than any stepwise process, such as multiplicative inverse of an infinitesimal. And the only way to come back is by a similarly powerful process. Also, there is a way to speak of ordinal succession in that remote zone, so we can reasonably discuss A = 1/m, or A = w + g.

So, as the causal chain back from the world we live in today to the remote past of origins is stepwise, it is inherently finite; there credibly was a beginning not a traversed infinite past.KFPS: HRUN, your attempted dilemma of leaving a dismissive barbed comment to stand unanswered or else be dismissed as putting up a meaningless “word salad” is duly noted for the ad hominem tactic and the underlying hostile attitude.

#93: You call it ‘ad hominem tactic’ while I call it predicting exactly what was going to happen.

And you are wrong, by the way, I did not suggest your argument should be dismissed because of a particular character attribute. Much rather, it is because your character I suggested to stop arguing. And your post in #93 shows that my judgment was right.

HR, doubling down, and trying to rewrite the record minutes later. Sadly typical. Kindly note, on substance, it is on the table again, that there is an endless number of counting numbers, which is why the set holds cardinality aleph null. In simple terms, endlessness of the first degree. We need a warp jump — e.g. reciprocal of an infinitesimal (remember the graph of y = 1/x from school?) — to get to that scale; not walking in steps no matter how long, which will always only reach to a finite number and will never exhaust the counting numbers. This, logically, applies to the successive casual chain back from where we are to the past of origins also. Hence, credibly, we have a finitely remote beginning, not an infinite past. KF

I repeat:

“He (KF) is right and you (DS) are wrong. And if math agrees with you (DS), then math is also wrong.”

HRUN, the issue on the table is to bridge step by step with finite steps from the finite to the transfinite and/or to bridge in the other direction by similar steps. That is an issue, a serious one; and one that it would be interesting to see your solution to. (Note, the remark that all natural numbers as defined, are finite, is a clue on just how serious the point is; I am perfectly willing to talk of ordinal, whole, counting numbers and to address such of transfinite vs finite scale.) FYI, in this general field of set theory and linked concerns a major backing up had to be had over what is now called naive set theory and in fact there is a significant dispute among relevant professionals regarding potential vs actualised infinities, e.g. cf here: http://projecteuclid.org/downl.....1203431978 . The matter is by no means as “simple” as dismissibly disagreeing with “mathematics.” KF

KF,

Thanks for the lesson on cardinal numbers. Now who was the one who claimed that any set with only finite elements is finite? Sorry, but it has to be said.

Ds, the cited is my reason for pointing out that the counting numbers 0, 1, 2 . . . stretch beyond the finite — the endlessness. The counting process is what is inherently finite and to use it as if it bridges to the transfinite to claim all natural numbers as defined are finite is where I have a concern. I repeat, as the causal chain back from the world we live in today to the remote past of origins is stepwise, it is ordered, countable in finite stages so is inherently finite overall; there credibly was a beginning not a traversed infinite pastKF

KF,

I have given up on the mathematical discussion, but you will need to formulate your arguments starting with these:

Peano Axioms

and their consequences. Statements about “causal chains to the remote past of origins” don’t get you anywhere.

DS, causal chains that regress stepwise to the past of origins are at the heart of the concerns. KF

KF,

Here is an example of what I’m talking about, courtesy of Terence Tao:

Short, to the point, no wasted words.

Tao doesn’t bring up anything about “causal chains back from the world we live in today to the remote past of origins”, nor would it have helped him. That’s irrelevant to the proof of the finitude of all natural numbers.

Are you guys sure you have any idea what you are arguing about? It seems to me that you are arguing entirely different things, and not paying attention to each other.

Maybe I’m wrong, but:

Dave says that all natural numbers are finite, but that the number of natural numbers is infinite, and given the name aleph-null.

kf says that you couldn’t start at “negative infinity” and ever get to any particular finite natural number in a finite number of steps.

Both of these seem like true facts to me, and neither person’s point bears on what the other person is saying.

What am I missing?

Aleta,

I think KF and I do understand each other. KF believes that transfinite natural numbers do exist, and therefore concludes that my clock example must in essence involve counting down to 0 from a transfinite number (which I also would agree is impossible).

I don’t believe in transfinite natural numbers, so I don’t think this is a valid objection.

I will add that in more recent posts, KF has alluded to “problems” with the definitions for the natural numbers I am using. It’s possible he may no longer claim that N contains infinite elements, but rather, they don’t accurately model the issue Spitzer is describing. I’m interested in the pure mathematics rather than pursuing this debate, though.

DS & Aleta:

The underlying context of concern is the claim that there is an infinite, completed causal stepwise succession from the deep past of origins to the present. That seems dubious on the ground that

any actualisable and completed finite stage stepwise process will be inherently finite. This, before we get to other issues on accumulations of entropy or the implied fine tuning of suggested earlier cycles of cosmic expansion, etc.In that context, it emerged that discussions of the set of counting numbers are fraught with issues on definitions and a set of proofs put forth that use counting processes to claim that while the set of natural numbers 0, 1, 2 . . . is endless [and of transfinite cardinality aleph null] all such numbers in the set are finite.

It seems to me that an inherently finite counting process cannot span the set, we need to reflect on the use of more powerful means that will get us to whole, counting, ordered numbers such as multiplicative inverses that start with the exceedingly small and catapult to the exceedingly large.

(As a simple finite example, the reciprocal of 10^-300 is 10^+300, 600 orders of magnitude away. The seemingly simple y = 1/x hyperbolic curve is fraught with powerful properties, including also being the root of the exponentials and natural logarithms. I am tempted to say here we see a mathematical wormhole, that for a short step into its mouth near 0 instantly catapults one far, far away. Taking the reciprocal of an infinitesimal, m, then can catapult us to a transfinite A. I have further specified that A will be a whole number value with the context that it is an ordinal, as the sequence from 0 up is just that. I then used the transfinite ordinals from w [–> omega, of order of magnitude aleph null] on to set a value A = w + g, g finite but large enough that no actualisable finite counting process can decrement it to zero. We can then look at w + (g – 1) and so forth, in succession as A ~1, A ~ 2 etc. A will be of order of magnitude aleph null. Likewise, we can then think of an increment from 0 as 1*, 2*, . . . n*, where in the context of a causal succession, n* will be now.)

I have an immediate concern with the definitions of the naturals and reals that would lock out various values or ranges, such as claiming that naturals are all finite, or that the real number line is in effect discontinuous exceedingly close to 0, i.e. there are infinitesimals that are close to 0 but are not real numbers, and the like.

That sounds far too close to being like a question-begging lockout of what is to be focal, and it leads me to see that there may be reasons why those who carried out non standard analysis spoke of hyper reals beyond the reals etc.

On grounds that the closed interval [0, 1] is generally acknowledged to be a continuum and part of the real numbers line, I suggest that there will be infinitesimals in the continuum, for example a convenient number m close to 0 and having useful properties. Such as, ability to have multiplicative inverse that will be a whole number with fractional part uniformly zero, and where also the values for such inverses of cases like m will fit on a sequence of numbers that is ordered in succession ranging from 0 up to and beyond w, etc. e.g. as clipped in 56 above:

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

In that context, we can examine the succession:

. . . A [= 1/m], A~1, A~2 . . . 2, 1, 0, 1*, 2* . . . n*

The leading ellipsis indicates that for argument, a prior endless succession is not ruled out to arrive at A.

The critical issue I see is to bridge either from 0 to A or else from A to 0.

On considering a stepwise succession of finite decrements from A, only a finite neighbourhood of A out to some A~s will be feasible, and indeed at any actually completed — key constraint — decrement of scope k, as the future is then open, there will be a further k + 1th step such that k is finite. Counting up or down is an inherently finite process so far as a completed count is concerned.

Posing a count that is not actually completed but only envisioned as 0 + 1 + 1 + . . . or the like is tantamount to saying we go on endlessly or to a transfinite number of steps

0 + 1 + 1 + . . . [transfinite number of steps of, order w] . . . 1 = w + r

i.e. such is to point to a potential but not actualised infinite process, which implicitly brings in the transfinite range again.

(My side note here is that it seems to me at this point, if the naturals are defined as an endless succession from 0, 1, 2 on and to have a transfinite total, there will be transfinites produced by such steps to get the transfinite overall cardinality. That is my concern on the claim that every natural is in fact finite while the set as a whole is of transfinite cardinality that DS has highlighted. I would be happy to acknowledge that every natural arrived at by an actualisiable and thus finite stepwise count is finite, but that is not the same as that there are no counting numbers that are not transfinite. if you want to call the collection of the finite counting numbers the naturals, it is then problematic, it seems to claim for it a transfinite overall cardinality. I would like to see something resolved about this concern but that is not crucial.)

The issue of counting up is likely to be straight forward: an actual successive count will be finite to the point reached but with an open future may continue.

For counting down, I would like to focus on the segment:

A [= 1/m], A~1, A~2 . . . 2, 1, 0

and to put it into a matching secondary sequence:

0#, 1#, 2# . . . s#

for every actually completed finite span to s#, there will only be a finite decrement. But a finite decrement from something of order aleph null will never get us to a finite range, we will still be in the transfinite zone.

Where the point of aleph null is that it denotes an endlessness, by contrast with an actualised actually completed count.

I argue further that an actualised completed decrementing count will be finite inherently and will not bridge from the transfinite to the finite.

Going further, a causal succession from the remote past of origins can be seen as subject to such a decrementing count to the singularity, followed by a finite upcount from the singularity to now.

Actually completing a count will therefore span back to some s beyond the singularity, which is finite. As in, not endless.

If a stepwise counting process has actually completed, then it is inherently finite and unable to span the gap to the transfinite or from the transfinite. If it is endless to allow spanning to or from the transfinite zone then arguably it is not completed nor is it capable of actual completion.Which is Spitzer’s point.

Again clipping:

So an exploration of surrounding mathematics points us to the force of Spitzer’s point, though it is likely to be much harder to follow.

The issue is conceptual, tied into the mathematical reasoning and to the import of what we bring to the table as underlying conceptions and commitments.

mathematics, now appears as a far ore human activity, not a decisive once for all battering ram of results claimed beyond correction or error. Such is of course also a message of Godel’s incompleteness results.

Proposing a world that comes to be out of non-being is not a credible start, we are forced to accept something as root of reality.

A proposed endless past causal succession runs into the issue that an actually completed stepwise process is inherently finite.

Credibly, the causal succession to the deep past of origins, is finite.

the root of reality on such a view, will be of a different order, a necessary being. One so anchored to the foundational framework of a world existing that once a world is actual it must be there. And not subject to contingent dependence on prior causes, particularly not a temporal succession.

Yes, this hints of eternal mind as root reality; something that reflects in evidences of design at cosmological level also. Where our finding ourselves inescapably under moral judgement, further points to such a mind as moral also.

Not a claimed proof, a way to see that such is not merely pointless and nonsensical speculation and imagination to be scoffed at.

Where, too often, such scoffing is a problem.

I must add, this thread has by and large been a genuine meeting of minds and a serious engagement of what is patently a thorny issue loaded with surprising connexions.

For instance, what happens when an algorithm goes infinite loop?

http://wayback.archive.org/web...../~csk/halt

(So also, are the advocates of an infinite past suggesting something comparable? Or is there an infinite loop driving an oscillator and feeding successive steps much as the clock signal in a computer processor physically, and can such be realisable for an endless duration?)

KF

DS, Seeing a message lead on main page but it will not load. KF

U/D: Seems the main page is stuck in an earlier state, weird.

KF,

If you have a calculus textbook handy, you likely will find in it a theorem which states that all rational functions (of which f(x) = 1/x is one) are continuous on their domains. This abrupt “catapulting” you describe is inconsistent with that theorem.

DS, Pardon but I am speaking of the nature of the function which moves

x –> [[ f(x) = 1/x ]] –> y = f(x).

Plug in the number x = 1*10^-300, and y will be 1*10^+300, 600 orders of magnitude away on a real number line. That is a very powerful case of shift in scale of values. Hence my remark on catapulting, or even comparison with the fictional wormhole that transfers one across LY in a blink. Move to the case of interest ultimately, an infinitesimal close to 0 for x and the output is transfinite. KF

KF,

Yes, I understand the function. It is a rational function with domain consisting of all real numbers except 0. It is continuous at every x except 0. This catapulting process would result in a discontinuity at a point other than 0, which is impossible.

DS, pardon but you are for a second time now setting up and knocking over a distortion of my point on the power of the function to move an input value to an output one, here with a 600 order of magnitude leap in size from x – 10^-300 to o/p y – 10^300, as has already been corrected. My point has exactly nothing to do with discontinuities in the hyperbola as it is all on the positive branch. A very small input value becomes a huge output value once manipulated by the inner mechanism. Imagine an op amp that does y = 1/x to see how remarkably powerful such a response is. KF

KF,

There is no catapulting of “infinitesimals” to transfinite numbers here, which you referred to in #109.

In the other thread, there are no numbers “g some large finite [so still of the scale aleph null]” either.

This is becoming painfully tedious, so I will step back for a while. As I said, why don’t you present your argument to WLC or someone else more sympathetic to your worldview? Perhaps vjtorley, who has a degree in math?

And again, I’m specifically referring to your cardinality argument, where you talk about transfinite natural numbers, finite numbers “of order aleph-null”, and so on.