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An infinite past?


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

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. daveS
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 kairosfocus
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. daveS
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 kairosfocus
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.
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. daveS
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. kairosfocus
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:
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.
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/20150221161433/http://www.cgl.uwaterloo.ca/~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 kairosfocus
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. daveS
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. daveS
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
KF, Here is an example of what I'm talking about, courtesy of Terence Tao:
Clearly 0 is finite. Also, if n is finite, then clearly n++ is finite. Hence by Axiom 2.5, all natural numbers are finite.
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. daveS
DS, causal chains that regress stepwise to the past of origins are at the heart of the concerns. KF kairosfocus
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. daveS
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 kairosfocus
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.
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. daveS
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/download/pdf_1/euclid.rml/1203431978 . The matter is by no means as "simple" as dismissibly disagreeing with "mathematics." KF kairosfocus
I repeat: "He (KF) is right and you (DS) are wrong. And if math agrees with you (DS), then math is also wrong." hrun0815
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 kairosfocus
#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. hrun0815
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. KF PS: 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. kairosfocus
HR, you have moved from the topic to personalities. KF kairosfocus
And now wait the long word salad that confirms exactly what I said in #89 ... hrun0815
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 if math agrees with you, then math is also wrong.
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
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 kairosfocus
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:
The picture of a lowest transfinite sitting next to a finite accessible to zero, one step away, becomes questionable.
Yes! It's completely ridiculous. That's why your position is untenable. daveS
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 kairosfocus
DS, no you have not
Then kindly point out the flaw in my proof. Edit:
And the problem is that the definitions you are using are caught up in that issue.
Specifically, what is wrong with post #80? daveS
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 kairosfocus
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.
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. daveS
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