An infinite past?

_{kairosfocus

January 26, 2016

Atheism, Logic and Reason, Mathematics, Sciences and Theology, worldview}

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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

Comments

But you do agree that your conclusion is that therefore an infinite past is impossible - true?Aleta_{January 27, 2016
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Aleta: I assume you also are summarizing your understanding of Spitzer’s argument I didn't actually read Spitzer's argument.mike1962_{January 27, 2016
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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_{January 27, 2016
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F/N: Let's do some clipping from Spitzer:
The problematic character of infinite past time is revealed by a seemingly inescapable analytical contradiction in the very expression “infinite past time.” If one splits the expression into its two component parts: (1) “past time” and (2) “infinite,” and attempts to find a common conceptual base which can apply to both terms (much like a lowest common denominator can apply to two different denominators in two fractions), one can immediately detect contradictory features. One such common conceptual base is the idea of “occurrence,” another, the idea of “achievement,” and still another, the idea of “actualizability.” Let us begin with the expression “past time.” Past time can only be viewed as having occurred, or having been achieved, or having been actualized; otherwise, it would be analytically indistinguishable from present time and future time . . . . Now, 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. No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable. Now, if “infinity” is applied to a continuous succession, and it is to be kept analytically distinct from (indeed, contrary to) “finitude,” then “infinity” must always be more than can ever occur, be achieved, or be actualized through a continuous succession (“one step at a time” succession). Therefore, infinity would have to be unoccurrable, unachievable, and unactualizable when applied to a continuous succession . . . . it might be easier to detect the unachievability of an infinite series when one views an infinite succession as having a beginning point without an ending point, for if a series has no end, then, a priori, it can never be achieved. However, when one looks at the infinite series as having an ending point but no beginning point (as with infinite past time reaching the present), one is tempted to think that the presence of the ending point must signify achievement, and, therefore, the infinite series was achieved. This conjecture does not avoid the contradiction of “infinite past time” being “an achieved unachievable.” It simply manifests a failure of our imagination. Since we conjecture that the ending point has been reached, we think that an infinite number of steps has really been traversed, but this does not help, because we are still contending that unachievability has been achieved, and are therefore still asserting an analytical contradiction.
I hope that helps. KF PS: Note his use of steps implies finite discrete stages in cumulative succession.kairosfocus_{January 27, 2016
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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?Aleta_{January 27, 2016
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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_{January 27, 2016
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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"Jack Jones_{January 27, 2016
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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.Aleta_{January 27, 2016
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Aleta, the core is a couple of paras, as I just linked. KFkairosfocus_{January 27, 2016
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But have I correctly summarized Spitzer's argument?Aleta_{January 27, 2016
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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.php?title=Mathematics#An_Analytical_Contradiction_in_.22Infinite_Past_Time.22 the issue is to traverse an infinite span in finite steps. KFkairosfocus_{January 27, 2016
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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:
Spitzer’s argument that it is impossible to traverse an infinite past to arrive at the present.
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_{January 27, 2016
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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.redwave_{January 27, 2016
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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).daveS_{January 27, 2016
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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:
I think the best — least prone to confusion — answer to the matter is that
if ascent to the infinite in discrete steps is a supertask (and it credibly is so), descent from the same . . . the transfinite zone or order of magnitudes (and particularly the suggested infinite past) . . . must span the same distance in the same sort of succession and would be highly dubious.

KFkairosfocus_{January 27, 2016
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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
I would be inclined to think that the transfinite character of a set or number should be characterised more by whether one can reach (or exceed) its scale by successive counting steps than by discussion as to what may or may not be of higher scale.
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
if ascent to the infinite in discrete steps is a supertask (and it credibly is so), descent from the same . . . the transfinite zone or order of magnitudes (and particularly the suggested infinite past) . . . must span the same distance in the same sort of succession and would be highly dubious.
You are also right to point to the absence of observational warrant for multiverse claims. KFkairosfocus_{January 26, 2016
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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.kairosfocus_{January 26, 2016
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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.jimmontg_{January 26, 2016
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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.daveS_{January 26, 2016
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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?daveS_{January 26, 2016
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I dunno about that. My mother's father repaired clocks, but I can't recall one that didn't require winding.Mung_{January 26, 2016
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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.kairosfocus_{January 26, 2016
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How many non-ticks between each tick, that's what I want to know.Mung_{January 26, 2016
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KF, Have you looked at the rules for cardinal arithmetic?daveS_{January 26, 2016
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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. KFkairosfocus_{January 26, 2016
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KF,
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. KF
Thanks for answering. Yes, I do claim that all natural numbers are finite. It's likely this the root cause of our disagreement.daveS_{January 26, 2016
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Phineas, the attempted bridge or rather cascade of bridges cannot be built. KFkairosfocus_{January 26, 2016
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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. KFkairosfocus_{January 26, 2016
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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.
DS, please. You know better, far better. If there are infinitely many naturals, there will be no last member, hence the ellipsis.
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).
And you obviously have not reckoned with why A is a transfinite integer.
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.
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.
No transfinitely remote real integers exist. Do you agree? Yes/No? Again, nonreal numbers are not on my time axis.
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
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?daveS_{January 26, 2016
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kf:
A transfinite span cannot be bridged in discrete finite steps.
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.Phinehas_{January 26, 2016
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