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

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:
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.
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. KFkairosfocus_{January 29, 2016
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UD Editors: Ginger Grant is no longer with us.Ginger Grant_{January 28, 2016
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KF,
DS, It looks to me that you have inadvertently begged the question at stake, traversing the transfinite in stepwise finite stages. KF
Can you tell me where this happened? Here are the steps:
1) Assume there exist infinite natural numbers 2) By the well-ordering principle, there is a smallest infinite natural number, which is clearly not 0. 3) If A is this smallest infinite natural number, then A - 1, which exists, is finite. 4) Therefore the infinite natural number A is the sum of A - 1, which is finite, and 1.
Where's the error?daveS_{January 28, 2016
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DS, It looks to me that you have inadvertently begged the question at stake, traversing the transfinite in stepwise finite stages. KFkairosfocus_{January 28, 2016
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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. KFkairosfocus_{January 28, 2016
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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_{January 28, 2016
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KF,
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.
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?daveS_{January 28, 2016
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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. KFkairosfocus_{January 28, 2016
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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 Grant_{January 28, 2016
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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. KFkairosfocus_{January 28, 2016
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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 can traverse the infinite in finite steps, namely: A, A - 1, A - 2, ..., 0. Just tack A onto the finite sequence above. Which is very ironic.daveS_{January 28, 2016
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KF,
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.
I agree that it has been ticking for a transfinite number of ticks.
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.
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.
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.
There are no real number infinitesimals.
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.
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 proof of 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 finite natural numbers. What is the cardinality of this set?daveS_{January 28, 2016
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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.kairosfocus_{January 28, 2016
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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. KFkairosfocus_{January 28, 2016
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KF, Just noticed this:
As to the clock does not tick at A, A is in the cardinality zone of the in principle countable transfinite aleph_null. KF
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.daveS_{January 28, 2016
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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.daveS_{January 28, 2016
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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. KFkairosfocus_{January 28, 2016
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I haven't really understood DaveS's point for quite a while, so I've not been thinking about that.Aleta_{January 28, 2016
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Aleta, 62 vs 63 is part of the context of 56. KFkairosfocus_{January 28, 2016
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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. KFkairosfocus_{January 28, 2016
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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 can count 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.daveS_{January 28, 2016
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Aleta, yes. He is saying there is an analytical clash between a finite process and a transfinite span. KFkairosfocus_{January 28, 2016
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kf writes,
Aleta, his focus is not on [whether] we cannot or do not reach the present
I think a word was missing, so I inserted what I think you meant.
but on there being an analytic contradiction between positing an inherently finite process and its claimed achieved traversal of a transfinite span
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 why we could not reach the present. I think I understand everything I need to understand now.Aleta_{January 28, 2016
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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. KFkairosfocus_{January 28, 2016
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In response to 56, kf, at least for me, all that continued explanation is unnecessary. I understand the argument.Aleta_{January 28, 2016
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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. KFkairosfocus_{January 28, 2016
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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:
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by labelling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange infinite collections of objects in order. Ordinals are distinct from cardinal numbers, which are useful for saying how many objects are in a collection. Although the distinction between ordinals and cardinals is not always apparent in finite sets (one can go from one to the other just by counting labels), different infinite ordinals can describe the same cardinal (see Hilbert's grand hotel). Ordinals were introduced by Georg Cantor in 1883[1] to accommodate infinite sequences and to classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[2] Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is w [--> omega], which is identified with the cardinal number aleph_0. However, in the transfinite case, beyond w, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely aleph_0 itself, there are uncountably many countably infinite ordinals, namely w, w + 1, w + 2, …, w·2, w·2 + 1, …, w^2, …, w^3, …, w^w, …, w^[w^w], …, E_0, …. Here addition and multiplication are not commutative: in particular 1 + w is w rather than w + 1 and likewise, 2·w is w rather than w·2. The set of all countable ordinals constitutes the first uncountable ordinal w_1, which is identified with the cardinal aleph_1 (next cardinal after aleph_0). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals to cardinals . . .
This at least looks promising, as it clearly points to whole numbers of transfinite nature, and distinctly identifies increments by addition to the next ordinal. Where cardinality at transfinite scale is an index of order of magnitude expressed at aleph null level by one to one correspondence. The logical next step is to suggest some finite counting number g, to be added to w, and put up as a further construction of A: 1/m = A (That is A * m = 1, multiplicative inverse. Where, m is an infinitesimal.) A = w + g In this context A less 1 would be w + (g - 1) . . . let us symbolise as A ~ 1, and so forth. Under these circumstances, it seems to me for the moment that A would be a transfinite not actually reachable from 0 by an inherently finite step by step process but is a whole number in an identifiable sequence. Reversing the matter let us now look at: . . . A, A ~ 1, A ~ 2, . . . 2, 1, 0, 1*, 2*, . . . n* A is obviously not a first step, the leading ellipsis takes care of that. For all we know for the moment an indefinitely large descending sequence has arrived at A. At least, we must be open to it. But, now we go beyond A and can make a correspondence of onward steps trying to descend to 0, say to be tagged with the singularity: A, A ~ 1, A ~ 2, . . . 2, 1, 0 . . . n* 0#, 1#, 2# . . . We face an inherently finite state based descent that can only ever be completed to a finite extent. But the span to be traversed to 0 is transfinite. 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. KFkairosfocus_{January 28, 2016
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kf writes, "I think, partly correct, partly not on Spitzer’s focal point" Good. kf writes,
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.
Again good, and in agreement, I think, with what I said. kf writes,
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.
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.Aleta_{January 28, 2016
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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:
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.
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:
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.
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.kairosfocus_{January 27, 2016
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Yes. The idea is logically self-canceling.mike1962_{January 27, 2016
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