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# Fun with the hyperreal numbers (and with the idea of an infinite actual past)

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The hyperreals are an extension of the real number line that brings to bear a reciprocal relationship between the very large and the very small. By so introducing extensions to the real number continuum, it forms a base for an infinitesimals approach to the calculus and makes sense of a lot of the tricks used by early pioneers of Calculus from Leibniz and Newton to Euler and beyond. (Though, it is clear in retrospect that they missed a lot of the pathologies that are now part of the far more cautious approaches of today.)

And yes, here is a case where Wikipedia does some good (likely, in a context where there are few basement trolls capable of making a mess):

Let’s zoom in on the graphic, which illustrates the “hyperreal microscope” of *R:

Let us note the definitional relationship between the infinitesimal and the hyper-large:

1/ε  =  ω/1

Where also, a common principle used is that ε is so small that ε^2 = 0. Where, to see what that is suggesting consider that (1/10)^2 = 1/100, and (1/10^50)^2 = (1/10^100), i.e. squaring drastically reduces the scale of a very small number.

This is all quite interesting, and has been used to rehabilitate some of Euler’s work, e.g. here.

(This is also quite relevant to some of the “Math tricks” used by Physicists and Engineers. The reference to the hyperreals may be a way to rehabilitate some seemingly dubious tricks.)

The principle that ω is a number greater than any finite sum 1 + 1 + 1 + . . . + 1 implies that it is of order type at least comparable to the first transfinite ordinal. The inclusion of the further numbers such as ω/2 indicates a reference to the surreals, and something like root-7 times ω indicates an onward transfinite continuum. I do not at this juncture specifically identify this ω with the familiar first transfinite. (Perhaps someone cares to clarify?)

So, we may at least highlight the surreals, where the vertical bars indicate continua — note the place for “infinitesimals”:

All of this is interesting in itself, as numbers are the tools of ever so much analysis and we here enrich appreciation of our favourite tool-box. (I confess, this weekend was more spent with dynamic-stochastic general equilibria, linked rational expectations, questions on modern theories of growth and human capital, etc. All, with Garrison’s Austrian approach to macroeconomics lurking, and blending in issues of saturation and stagnation at points along the PPF as well as what happens to shocked economies with low investor confidence . . . as in, 20+ years on from devastating volcanic eruptions. This stuff was the oh, what about light exploration as a relief.)

But all of this converges on something which has come up for strong, sustained exchanges several times here at UD. Namely, the suggestion of an actual infinite causal-temporal past of the [wider?] cosmos. For, if ω is such that no finite succession from 0, 1, 2 via 1 + 1 + 1 . . . + 1 can reach it, then counting down — notice, the ladder-like succession of steps (and how the surreals extend this to construct continua and to go into transfinite ordinals) — from it in finite succession [or by symmetry counting algebraically upwards from – ω] may reach to something like ω/2 [or – ω/2] but it will be futile for getting to a finite reach of a zero-point.

In short, we can see here a reason to hold that there was no actually transfinite causal-temporal succession of states that have managed to reach the present. Nor will it do to posit that at any given past time p that can finitely succeed to now, infinity past was already traversed. That begs the question of HOW.

This also surfaces a logic of being point.

Namely, that non-being has no causal powers, so that if ever there were utter nothing, such would have forever obtained. Thus also, circular causation is forbidden as this would imply that the not yet existent acted as a cause. Thus, we either have an infinite succession of contingent beings as the world-root or else there is a necessary being at world root. That is, an entity such that it is utterly unlike a fire, which has several required external, enabling “on/off” causal factors for it to begin or be sustained:

Where this goes, is that a necessary being is framework for any possible world to exist. So, in any world, it would be. It neither began nor can it cease from being. For instance, in reality we may consider a world W, which must have a distinct identity. So, too, we may consider some distinct thing in W, A that contributes to its identity. Then, we may look at W = {A|~A}. This shows two unities, i.e. two-ness. No world is possible without two-ness, and beyond, the panoply of numbers.

This is what gives bite to Berlinsky’s remark that was just raised here at UD:

>>There is no argument against religion that is not also an argument against mathematics . . . .

Mathematicians are capable of grasping a world of objects that lies beyond space and time ….

… Come again …

DB: No need to come again: I got to where I was going the first time. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects.

… And this is something that you, a secular Jew, believe? …

DB: What a question! . . .  I have no religious convictions and no religious beliefs. What I do believe is that theology is no more an impossible achievement than mathematics. The same rational standards apply. Does the system make sense; does it explain something? Are there deep principles at work. Is it productive? >>

So, now, we see from the hyperreals augmented by ideas of causal-temporal succession, that it is hard to defend the notion of a transfinite actual past of contingent beings leading up to now. This points to there being an actual beginning of the world and that this traces to a finitely remote necessary being world root.

That’s enough for a UD Sunday reflection! END

PS: As it is being claimed or implied that no serious thinker thinks like that, I add a clip, just for record:

{} --> 0 {0} --> 1 {0,1} --> 2 {0,1,2} --> 3 . . . {0,1,2,3 . . . } --> w
5 An actual infinite cannot exist. [He has many reasons for that, which are often hotly contended but on the whole I think he has a serious point . . . at minimum he has shown that modern atheism effectively is reduced to a highly contentious assumption about the past, which is very different from its boast that it can be seen as a default for the intelligent person] 6 A beginningless temporal series of events is an actual infinite. [Note this] 7 Therefore, a beginningless temporal series of events cannot exist. Since (7) follows validly, if (5) and (6) are true the argument is sound . . . . >> 6.3 Successive Addition Cannot Form an Actual Infinite Craig’s second argument addresses this very point. 8 The temporal series of events is a collection formed by successive addition. 9 A collection formed by successive synthesis is not an actual infinite. 10 Therefore, the temporal series of events cannot be an actual infinite (Craig 1979: 103). The collection of historical events is formed by successively adding events, one following another. The events are not temporally simultaneous but occur over a period of time as the series continues to acquire new members. Even if an actual infinite were possible, it could not be realized by successive addition; in adding to the series, no matter how much this is done, even to infinity, the series remains finite and only potentially infinite. One can neither count to nor traverse the infinite [--> in such a stepwise fashion] (Craig and Sinclair 2009: 118). It might be objected that this sounds very much like Zeno’s paradoxes that prohibit Achilles or anyone from either beginning to cross an area or succeeding in doing so. But, notes Craig, significant disanalogies disallow this conclusion. For one, Zeno’s argument rests on progressively-narrowing, unequal distances that sum to a finite distance, whereas in traversing the past the equal distances continue to the infinity of the future. [–> relative to the distant past in view. This is also the context in which I have consistently spoken of finite stage causally successive stages] Second, Zeno’s distances are potential because of divisibility, whereas the distances from the past are actual distances or times to be traversed.
Beyond, lurks time's arrow; entropy. Part of the driving-force of causal-temporal succession is that there are rich concentrations of energy that support processes of change. But that means the sources are gradually degraded and as they are finite, we look at what has been called heat death. Such is a condition where the degradation of energy has attained a point where the energy sources -- stars are especially in view -- have been used up. That we are not in this degrades state implies, then, a finite date; at least i/l/o understanding the cosmos as a whole as an isolated system. Of course, it was suggested above, that perhaps there is an external non-physical source able to sustain the world "forever" through adding fresh energy. This is in effect continuous creation by an entity that would recognisably be God. And that is a shocker for confident evolutionary materialism advocates. Is there a physically beginningless world coeval with God who sustains it from without? To such a model, we can note that one may speculate as one pleases, the evidence we can observe prunes speculations by highlighting evidence of a beginning rooted in back-projection of an expanding world with background microwave energy consistent with the about 14 BY origin. Likewise, we would expect to see a huge number of degraded stars, including cooled down white dwarfs. These are not seen with any numbers to support a beginningless world working much as what we observe. In all of this light, the best explanation on the table is finitely remote origin of our world, tracing to a necessary being world-root. Beyond, we may discuss implications of such a world having in it conscious, minded creatures able to seek out explanations of an intelligible world, who find ourselves under moral government. Such is an onward exercise and the road to it lies through the AI issue, including the further exploration of memristors. DV, another day. KF kairosfocus
KF, Thanks to you as well. daveS
DS, I thank you for the discussion. However, it is still a question: does or does not a beginningless past with members of the actual past being beyond any particular stated n in N, imply a traversal of the transfinite? If it does, I suggest the transfinite traversal done stepwise is questionable, and I would say credibly a futility. Also, to suggest that for any particular R in the past the traversal from the beginningless past can be regarded as already complete, seems to me to be a skirting of the precise point at stake. So, while many do argue such a past, and while it is obviously so hard to discuss it with any reasonable hope of a resolution that I can see why many would not address it at all, that question is there on the table. KF kairosfocus
KF, May I suggest you contact David Snoke or WLC and present your ideas to one of them? At this point I don't have anything more to say. daveS
DS, what I have argued is that there is good reason to doubt claims that are sometimes made for a beginninless physical cosmos. Those reasons start with what is REALLY meant by such a claim. If the past counted by stages [e.g. years] extends limitlessly beyond any natural number n in N, then that is a claim to a past infinity. As the past is not a concept but actuality that has been succeeded by subsequent stages though causal processes, then it is natural to ask what would such an infinitely remote actual stage look like. The natural response would be to model such, and the hyperreals line gives us a way. The dynamic of succession indicates that a stepwise incremental process cannot span from such a past to now. The answer is, no, what is meant is that every actual past value is finite but the overall chain extends limitlessly beyond any finite counting number n. That is at least close to borderline incoherent. The response is, putting conceptual braces around the chain does not give it a duration that extends beyond that of actual past stages. The above is perilously close to implying such stages of order w. No, they are all finite but extend beyond any particular finite counting number n, no matter how large. On one side, that suggests at first an indefinitely large but finite duration. But no, it must go beyond any such ordinal . . . and a successive count to the past is a chain of ordinals. The order type of that chain is of course w, a transfinite ordinal. But then, such a suggestion is before actual proofs: 0,1, 2 ... w . . . The ellipsis of limitlessness in the context of counting numbers points to that order type. But, maybe it is just a recognition of endlessness of succession appended to the naturals for convenience? The other approach, the hyperreals unifies the two, no it is not simply recognition of a new class of quantity, it is legitimate to see the number line extended to hyperreals. So, I say, values q of order w exist as numbers, the question is, can they exist as past times. If no, then the ellipsis needs to be explained in terms of what its order type means as applied to the actual temporal order. Leaving that vague as every case is finite but the cases go on and on beyond any n in N sounds less than satisfactory. It sounds suspiciously close to exploiting the grey zone between concepts to have one's cake and eat it. And that is before the thermodynamic limitations come in play. Where, if your intent is to address the Kalam cosmological argument (in which, oddly I have little interest) then invoking God as stabilising thermodynamics seems to yield the wider point. On cosmological reasoning I would suggest that the existence of a contingent world suggests dependence on a higher order of being. I don't think your objection is a serious one. Back to WLC, let me again note from the SEP summary, clipping 35 above, also addressed to you:
5 An actual infinite cannot exist. [He has many reasons for that, which are often hotly contended but on the whole I think he has a serious point . . . at minimum he has shown that modern atheism effectively is reduced to a highly contentious assumption about the past, which is very different from its boast that it can be seen as a default for the intelligent person] 6 A beginningless temporal series of events is an actual infinite. [Note this] 7 Therefore, a beginningless temporal series of events cannot exist. Since (7) follows validly, if (5) and (6) are true the argument is sound . . . . >> 6.3 Successive Addition Cannot Form an Actual Infinite Craig’s second argument addresses this very point. 8 The temporal series of events is a collection formed by successive addition. 9 A collection formed by successive synthesis is not an actual infinite. 10 Therefore, the temporal series of events cannot be an actual infinite (Craig 1979: 103). The collection of historical events is formed by successively adding events, one following another. The events are not temporally simultaneous but occur over a period of time as the series continues to acquire new members. Even if an actual infinite were possible, it could not be realized by successive addition; in adding to the series, no matter how much this is done, even to infinity, the series remains finite and only potentially infinite. One can neither count to nor traverse the infinite (Craig and Sinclair 2009: 118). It might be objected that this sounds very much like Zeno’s paradoxes that prohibit Achilles or anyone from either beginning to cross an area or succeeding in doing so. But, notes Craig, significant disanalogies disallow this conclusion. For one, Zeno’s argument rests on progressively-narrowing, unequal distances that sum to a finite distance, whereas in traversing the past the equal distances continue to the infinity of the future. [–> relative to the distant past in view. This is also the context in which I have consistently spoken of finite stage causally successive stages] Second, Zeno’s distances are potential because of divisibility, whereas the distances from the past are actual distances or times to be traversed.
In short WLC is raising arguments that should sound at least vaguely familiar. If one explicitly argues that the l-ward ellipsis points to actual past moments q or order type w or leaves the matter in the vague terms of oh the l-ward succession is beginningless and has members that exceed any specific n in N no matter how large, it seems that you are trying to traverse the transfinite in finite stage steps. A futility. KF kairosfocus
KF,
And if the chain is infinite, for argument, then it is because particular events have been so far superseded by the next then the next etc that they are now remote beyond any arbitrarily high stage count.
If by this, you mean just that there is no maximum separation between the present and past events/stages, yes, I agree. If you mean there is a particular past event/stage separated from the present by infinitely many steps, then I would disagree.
What is a lot less clear is whether the murky zone l-ward ellipsis of the reduced succession you keep on truncating to is a way to have q without acknowledging it. Likewise, we do not get to impose what is convenient to our worldview preferences by default on logical possibility in our view is enough to effectively infer truth or actuality. Comparative difficulties on factual adequacy, coherence and explanatory power apply.
From my perspective, you are the one adding these infinitely distant q-points to the standard model of an infinite past without justification. The standard model which theists such as WLC and David Snoke use, incidentally. In order for me to take this model with q-points seriously, you would need to provide a rigorous proof that they must exist, that is, that IP(−ω) implies IP(+ω). I do acknowledge that you have attempted arguments in this direction, but AFAICS, they have been non-sequiturs.
Just what is a beginningless, beyond any specific n in N span in the past if it does not contain any actually transfinite members of the class w etc? Does that make a material difference?
It seems to be, for both of us. You have argued strenuously that it is nonsensical to call such a past "infinite". I have pointed out that it would be most convenient to use definitions consistent with that of WLC and the rest of the published literature on the subject, so we can at least be clear on what those authors are saying.
On reflection, the restriction to finitely large values is implicitly shaped by the physical consideration of actual stepwise succession of stages. But, physical considerations cannot be cherry-picked, feasibility requires that the implied energy dissipation in causal-temporal succession also applies. So, an actually transfinite span — murky or not — cannot be actually bridged and were such to happen, energy in the cosmos would have so degraded that life such as we experience would be physically impossible.
Recall that I am essentially working in a theistic perspective here, as is WLC. Ultimately, I am challenging his Kalam argument. Now recall some common theistic assumptions: God created the universe, God exists in some timeless realm and can view all of our time at once, and very importantly, God intervenes occasionally our universe. Surely under those assumptions, God could step in every few billion years and "turn back the entropy clock". daveS
DS, I have been busy, let me turn to something key. I see you in 90:
{KF:] This also means that the set of the past events and stages cannot have duration beyond that of these events or stages. [DS:] I don’t agree with this principle. How would you apply it to a set of past events where there are events at arbitrary finite distance from the present?
It is central to our considerations to recognise that we are dealing with the real world, not a mathematical thought space. Collecting the chain of past events or stages into a set by in effect putting conceptual braces around them CANNOT change the actual stages of the past. And duration to now for a past stage R is - R_t. That is, duration since attaches to the actual stage of the past. As a direct consequence, the duration of the actual past as a whole similarly attaches to the chain of events. Surely, it is possible for that chain to be finite, in which case the overall duration of the chain is Max ( - R_t), i.e. the duration since the first event. And if the chain is infinite, for argument, then it is because particular events have been so far superseded by the next then the next etc that they are now remote beyond any arbitrarily high stage count. I have tried to capture such an event, Q, by using models of the number line that use hyperreals tied to infinitesimals, using reciprocals and the hyperbolic function to catapult to such a range. This, because you have spoken to beginninglessness again and again. For example:
Beginningless so transfinitely remote range, where on actuality, q is a member: . . . q –> q+1 –> q+2 –> . . . procceeding stepwise to a finite range down to now: –> p –> p+1 . . . –> 0
Reducing and putting braces to highlight the set of the actual past, S: S = {. . . q –> q+1 –> q+2 –> . . . –> p –> p+1 . . . –> 0} This, you have objected to, in effect trying to impose that the values are all finite but go beyond any arbitrarily high specific finite value: S = {. . . q –> q+1 –> q+2 –> . . . –> p –> p+1 . . . –> 0} Ironically, it is exactly the issue of finitude of stepwise R-ward succession that has been central to my point: e.g. stepwise cumulative succession of finite steps from a reference point 0 may amount to an arbitrarily large value k, but such a value will then be bounded and exceeded by k+1, k+2 etc. Thus, I have emphasised the impotence of stepwise succession to actually span a transfinite range, to actually reach the infinite. In that context I have consistently pointed to the ellipsis or with lines the pointing arrowhead. Sometimes, "etc." fills this role. Stepwise succession may point beyond itself to the potentially infinite, but it cannot span to that zone that is so easily reached by the hyperbolic catapult from infinitesimals in the continuuous interval (0,1] . . . where continuity is key and helps us appreciate that e such that e^2 ~ 0 is in the line in the near-neighbourhood of 0 where taking reciprocals will fail at 0 itself. This is how we reconcile speaking of hyperreals beyond any finite counting number k attainable by stepwise succession from 0. Now, time acts causally and cumulatively, R-ward in the sequence. In speaking of beginninglessness, you are focussing on an ellipsis L-ward, and wish to confine it to speak only of an indefinite, limitless L-ward extension of predecessor finite values. The root challenge here is tied to how we get the L-ward side of a number line, multiply through the R-ward side by - 1. We thus can reflect on how a succession from some j to j+1 etc and onwards to k can be exceeded by k+1, k+2 etc on. That is, stepwise succession cannot actually span the limitless, beyond any arbitrarily high but finite span [k - j]. Where, multiplying through by - 1 so to speak, or merely reversing direction cannot change the underlying dynamic of only attaining to a finite span of succession. In this context, playing around with ellipses that are now finite then transfinite then finite again to suit points becomes fatally equivocating. For, there is no clear dynamical means to span a limitless range. And, it therefore makes sense to speak of some q in that far past zone which is so far succeeded that it is limitlessly, transfinitely remote. Such a q cannot be succeeded stepwise to 0, now. That is clear. What is a lot less clear is whether the murky zone l-ward ellipsis of the reduced succession you keep on truncating to is a way to have q without acknowledging it. Likewise, we do not get to impose what is convenient to our worldview preferences by default on logical possibility in our view is enough to effectively infer truth or actuality. Comparative difficulties on factual adequacy, coherence and explanatory power apply. Just what is a beginningless, beyond any specific n in N span in the past if it does not contain any actually transfinite members of the class w etc? Does that make a material difference? It seems to me extraordinarily like a have your cake and eat it rhetorical situation. Especially, where the same causal dynamics by which the present stage becomes the next have lurking in them the inexorable degradation of energy availability to do work. And, for feasibility, all materially relevant conditions must be fulfilled. Indeed, entropy has been spoken of as time's arrow. An actually beginningless, ultraremote past for a cosmos that as a whole constitutes an isolated thermodynamic system will -- in succeeding itself stepwise to now -- have so degraded its available energy by now that the sort of life we enjoy would not be possible. IIRC 10^25 s is a yardstick of such time, about 100 mn times the duration since the big bang. Large but definitely finite and well within 10^100 s. Of course as Sears and Salinger remarked in my foundational thermodynamics textbook, that the physical cosmos as a whole is isolated is an open question. That is, a quasi-infinite physical world would be contingent on an external enabling entity. One, extraordinarily like God, if one is interested in cosmological arguments. (My actual focus is on the physical import of the temporal-causal successiveness we note, as we look back to origins.) A world of actual physical reality makes the suggestion of a beginningless past highly questionable, on grounds tied to causal-temporal successiveness. Even, if one can eliminate q definitively and avoid effectively putting it back in under another symbol, j. Back on that, it seems to me that stage R taken as a pure counting number r, will span all ordinals in the L-ward sense. That leaves very much open that some would be beyond w. With the hyperbolic catapult and -1 multiplier mirrors in play, there is no inherent reason that I see to cut off cases where r is in a zone where the yardstick -(1/e) = - (w/1) does not apply. That is, number line and REAL number line are different, the latter being a subset. Where a function based on the interval (0,1] is able to generate the wider span. On reflection, the restriction to finitely large values is implicitly shaped by the physical consideration of actual stepwise succession of stages. But, physical considerations cannot be cherry-picked, feasibility requires that the implied energy dissipation in causal-temporal succession also applies. So, an actually transfinite span -- murky or not -- cannot be actually bridged and were such to happen, energy in the cosmos would have so degraded that life such as we experience would be physically impossible. More to the point, an oscillating cosmos type model would dissipate after ~ 100 bounces and the evidence seems to be that we are having accelerating expansion rather than the slowing down that marks moving to an extreme of an oscillation. A budding-off or bubbling model would run out of available energy. Multiverses popping up out of utter non-being are not plausible. Circular causation is a species of popping out of nothing. All of this leaves as best explanation, finitely remote origin from a necessary being world-root. (Indeed, we are actually debating the nature of such a NB, and my note on energy dissipation is a way of saying the physical cosmos -- even if extended to a beginningless past -- is radically contingent so not a plausible NB.) KF kairosfocus
PS:
So, we are invited to believe in a beginningless, beyond limits, already traversed, succession at any particular stage. One where between any two particular states the duration is finite, but the whole is held to be infinite, beyond any finite span of stages. There are fairly obvious tensions here.
I will point out that even if we call such a past "finite", in such a scenario the universe would not have begun finitely many years/stages ago. Therefore in order to get the Kalam Cosmological argument to work, one would need to show this sort of "finite" past is impossible. daveS
KF,
I find that you are in effect saying that the infinite span of steps — the main thing to be explained — has already happened at any given R leaves the key question on the table as though it has no significance.
I'm not claiming it has no significance. Keep in mind I'm not trying to "explain" how all this came about. I'm in effect asking whether you can show all this is impossible.
This also means that the set of the past events and stages cannot have duration beyond that of these events or stages.
I don't agree with this principle. How would you apply it to a set of past events where there are events at arbitrary finite distance from the present? That is, where there are events 1 step before present (BP), 100 steps BP, 10^10 steps BP, 1 googol^googol steps BP, and in fact at any positive integer number of steps BP. What is the duration of this particular past? It's clearly not the duration between any particular event and the present. But if it's finite, it must have some definite value, correct? What is that value, and how do you make that choice?
So, we are invited to believe in a beginningless, beyond limits, already traversed, succession at any particular stage. One where between any two particular states the duration is finite, but the whole is held to be infinite, beyond any finite span of stages. There are fairly obvious tensions here.
That is an accurate description of the scenario, of course, but I am not detecting any tension. daveS
jdk, I do agree with everything you said in #86. I think that's the "standard" model of time that practically everyone uses, and that while timepoints separated by infinite distances are conceivable, they are not physically realistic. daveS
JDK, on the claim of an actualised past infinity, I am by no means convinced that we are speaking of the reals only, that needs to be shown -- especially as the duration since the past is a duration since specific actual events or stages. You may not be interested in the claimed infinite past but that claim is interested in you, on the roots of the world around us. Your projection of a future as infinite fails to reckon with the difference between a potential and an actual infinite. That is, from any given now, the future is open-ended but cumulative. Likewise, you have simply restated that the past is infinite, begging the question at stake, especially given the dynamic at work, stepwise succession of -- for convenience -- years. The key thing that lends limitless character to the negative number line is the ellipsis or the pointing arrow, indicating as a concept limitless continuation . . . no-one has argued that there is a single point of transition to the infinite n + 1 = "infinity," it is the property of limitlessness that leads to recognition of a new type of quantity, the transfinite. However, the analysis on taking reciprocals from (0,1] as the values tend to 0 above goes beyond that which additive succession gives us, allowing us to see how there is a reasonable continuity of range though of course there is that ellipsis there to indicate there is a span of transfinite character involved. That is, the hyperreals perspective allows us to see that as we tend to an e such that e^2 ~ 0 we are in a relevant range that is transfinite for 1/e = w/1, though of course there is continuity right down to zero on the interval implying a fundamental unity to the whole process. By contrast successive addition only attains the potential infinite, as it has weaker power. The ellipsis gathers up "the rest" in one conceptual leap. And as the negative reals are in effect the mirror image if we know that we cannot span up to the actual infinite by successive addition, there is reason to see that claiming to descend form the same scale is not going to be feasible, where claiming oh it has already happened at any given R seems very evasive. But stepwise succession of transfinite character is precisely the thing to be grounded on a reasonable dynamic of causal-temporal succession, where there is reason to see that such a succession cannot span a transfinite. I note, circular causation in a chain implies that a future, non existent entity causes itself, and that descent from infinity in steps is questionable, leaving the only serious option, a finitely remote world-root. This last is also supported by the point that the entropy of an infinitely old world should have long since reached to utter degradation. KF kairosfocus
DS, I made several responses in the range 60 - 7 and am surprised that you do not find them clear enough. I summarise, that the thing to be explained is the claimed infinite past, i/l/o the context of a known step by step dynamic that cannot span an infinite range. I find that you are in effect saying that the infinite span of steps -- the main thing to be explained -- has already happened at any given R leaves the key question on the table as though it has no significance. Recall, the actual past is such that every past state was once the present, then was succeeded by the next and so forth. This also means that the set of the past events and stages cannot have duration beyond that of these events or stages. Putting them inside a set of braces adds nothing, it is the ellipsis of limitlessness that is key and what it here implies is an assumed completed infinite succession already completed. So, we are invited to believe in a beginningless, beyond limits, already traversed, succession at any particular stage. One where between any two particular states the duration is finite, but the whole is held to be infinite, beyond any finite span of stages. There are fairly obvious tensions here. You may believe that an infinite succession of the actual past has already occurred at any specific point, that that infinite succession was somehow spanned step by step, that every stage in that infinity is only finitely remote from us but the whole is beyond all limits in the past -- where finite directly implies within a limit, and so forth, but I think something is wrong with the cluster of claims. KF kairosfocus
So Dave. What do you think of this formulation: Let the number line be a model for time, with an arbitrary point P0 the present, and the future in the positive direction and the past in the negative. For convenience sake, consider the integers only (with the reals to be understood as "filling in" the rest of the line), so that there is a discrete distance of 1 unit between "moments". For any positive point F in the future, F + 1 is even further in the future. Therefore, the future is infinite in the sense of being unbounded or unlimited, even though every point in the future is a finite distance from the present. There is no point in the future that we can't eventually "get to" from the present, even though there is no limit to how far in the future we might go. Likewise, for any negative point P in the past, P – 1 is even further in the past. Therefore, the past is infinite in the sense of being unbounded or unlimited, even though every point in the past is a finite distance from the present. There is no point in the past from which we can't "have gotten to" the present from, even though there is no limit to how far back in the past we might go. The confusion here is that "infinite" is not a place on the number line, or a property of a particular point: you can't be "infinitely far away" from the present. "Infinite" refers to a property of a sequence of numbers, stating that further numbers can be generated without end. jdk
jdk, I'm not the thread owner, but these related points are of interest to me as well. I think there is essentially no chance of agreement on the specific question in the OP, but the most interesting parts of these threads has been the side discussions that arise. daveS
Hi Dave: my apologies for busting in, as the main topic here, between you and kf, is whether an infinite past is possible. (I think that's the topic.) The fact that the the distance between any two points is finite appears to be accepted by both of you. So I'll bow out, and let kf think about your 83. jdk
KF,
Last, that any actual separation of any two distinct stages in time will be finite is another way of saying the past is finite.
I say this is false. Now how do we resolve the dispute? daveS
kf writes,
JDK, transfinite separation on a number line is actually meaningful, cf. OP
Not if we are just talking about the reals (or the integers) on a number line. kf writes,
More relevant to our concerns ...
, but he means "More relevant to my concerns ...", as I am not interested in the issue of an infinite past. jdk
JDK, transfinite separation on a number line is actually meaningful, cf. OP. More relevant to our concerns, if any two stages of the actual past will be finitely separated, then we have a fairly good reason to hold that it, as a whole, is therefore finite. Just set one as now [ at 0], the duration since any other stage R, -R_t will be finite. Conceptually collecting the causal-temporal sequence of stages into a set cannot add any further duration to them. But of course, what is being pushed in is the ellipsis, indicating that beyond any particular R, there are limitlessly more prior stages, beyond any particular natural number, regardless of how large. Thus, we are invited to accept that there has been an infinite causal-temporal succession of past stages (years, for convenience), all of which are somehow also finitely remote from now. Further, we are invited to accept that the limitless succession is always already accomplished when we contemplate a given point or stage, so that succession from some specific past point R to now is of no consequence. My first concern is that no causal-temporal stepwise, finite stage succession can traverse an actually transfinite number of stages, for fairly obvious reasons. After k stages we may go on to k+1, k+2, etc. So, we are as good as at the beginning, 0. But, it will be suggested: at any particular stage, there is no beginning, the limitless succession has already happened. Already, a problem, the big event is off stage, invisible, unobservable and inexplicable. Yet, it is the key to the whole. Big questions lie unanswered, even unanswerable on the scheme of thought. There is even a suggestion, to ask for an intelligible, reasonably empirically or analytically anchored explanation is to resort to the much deprecated principle of sufficient reason. However, to inquire into why is not to presume an answer, though we have a known dynamic of succession at work and given any present state -- it needs not be initial -- a transfinite traverse of stages cannot be completed, only a finite number. Recall, the actual past is such that every past state was once the present, then was succeeded by the next and so forth. So, we are invited to believe in a beginningless, beyond limits, already traversed, succession at any particular stage. One where between any two particular states the duration is finite, but the whole is infinite, beyond any finite span of stages. There are fairly obvious tensions here. But of course, circular cause and appearance from nothing are appeals to cause from non-being. A finitely remote beginning requires a necessary being world root, which to many is unacceptable for essentially worldview reasons. That leaves the above or something much like it as the view to be stoutly held. That set of alternatives speaks. KF kairosfocus
No, kf, I'm not "missing the issue": I am just not interesting in, nor addressing, the issues your are discussing about time and the nature of the past. I am merely saying that, to use your words, "any actual separation of any two distinct stages in time will be finite", and I agree with you about that, and you agree with me that "there are no two points that are “an infinite distance” apart." That's all. However, that also means that the phrase "transfinitely remote" in respect to points on a number line is meaningless. jdk
JDK, it seems you keep missing the issue. I am not defending that our actual world timeline can have an infinite, beginningless past. I am showing that such a claim implies that there are actual past stages that were once the present but have been succeeded stage by finite stage to now be transfinitely remote. I have put on the table ways to represent that situation, and we can see that they are inconsistent with the dynamic of stepwise, finite stage succession. The idea that oh we can assert that at any past stage R, there always were limitless further past stages (think, years for convenience) -- notice, the telling beyond limits -- amounts to saying that we have a cumulative transfinite past, as the set of past events cannot have a duration beyond the particular events in it. If it is transfinite, there are stages in it that are at transfinite remove. The set only adds conceptual braces around the actual past stages, it does not manufacture additional duration out of the mere act of collection. Last, that any actual separation of any two distinct stages in time will be finite is another way of saying the past is finite. KF kairosfocus
KF,
If we are in the beyond counting and bounding by successor range we are in the hyperreal transfinite domain.
But we aren't in the "beyond counting and bounding by successor range". All the proposed points in time/stages are finitely many years/seconds/steps from the present. daveS
KF, I'm not trying to "lock" anyone up here, certainly. However, you made a request in your post #62, which I responded to in #65. Isn't it fair to ask you whether my post #65 is a satisfactory response to your request? I simply cannot tell from your #70. daveS
When I wrote, "Given a line of infinite length, there are no two points that are “an infinite distance” apart.", kf replied
a line of infinite length is not a physically instantiated reality in an actual space, it is a concept.
This is true. There is nothing that we know of that is of infinite length in any actual space. When talking about infinity, I don't see how we can be doing anything other than talking about concepts. Also, kf writes,
Thus to specify or even symbolise particular points will always yield a finite separation
I take it therefore that kf agrees with my original statement. All the rest of the concepts he offers about hyperreals etc. are additional thoughts and topics, but they don't change the fact that there are no two points that are “an infinite distance” apart." jdk
DS, I will not play the yes/no interrogator lockup game. I have repeatedly and with significant discussions pointed out why I am unhappy with the constructions that have been put. I am currently expanding on the concepts involved as just now. I have reason to hold my views and have my concerns, they are not just irrational or stupid, incoherent, stubborn rants -- the force of some dismissive language above is all too plain. By dint of right of reply, I already have reason to know that my intuitions and explorations of was it two years ago were given inappropriate short shrift, and I think I have a right to draw them out in these current reflections. Maybe I will see that there are needed changes, but so far it looks to me like the power of the ellipsis beyond limits needs to be clarified. It looks a lot like, a second approach to the transfinite through reciprocals is fruitful and shows ways to fill in what is not communicated by successive addition. It also bridges to issues of coherence in the Calculus in terms of learning and forming concepts. KF kairosfocus
DS, Let us follow up that definition. Can we with profit pre-assign a transfinite number and view it as a reasonable extension to the naturals and reals? I believe so, let me clip from 63 above:
an infinite line as proposed is such that it corresponds to the reals, for which endlessness on either side of 0 is material. That is, for any specified point along the line, there is more beyond without limit. Thus to specify or even symbolise particular points will always yield a finite separation as the beyond implies a bound, but that does not tell the whole story that is expressed in the ellipsis in many contexts or by pointing arrows in the case of lines. Indeed, it is endless continuation which is the heart of our understanding of the infinitude of the reals or the integers. In this context, I point to the OP above to the Hyperreals and infinitesimals as giving an alternative way to conceptualise using the catapult power of the reciprocal function thus yielding a continuum that enfolds a span of numbers that when — using w for omega and e for epsilon — we take 1/e we have w, and the converse. Where e^2 ~ 0 [ADDING: , so also [2e]^2 ~ 4 x 0 --> 0, i.e. we can see a stepping down in this zone to 1/2e = 1/2 x 1/e = w/2 etc and will recognise that e is a yardstick for a zone that also extends above it significantly.] That is there is a cloud of all but zero numbers that allows us to create a conception that is relevant. I add: if you will, pivot discussion on two operations, + and x with identity elements 0 and 1 used as start-points. From 0 by successive addition we see unlimited incrementation beyond, leading to a view of the infinite. Namely, we see an endless successiveness and infer to a new order of quantity, order type of such endless succession, a transfinite result. Of course I have in mind here the von Neumann succession, and we can fill in a continuum by using in effect power series based on 10^-n or the like base, decimal fractions etc continued endlessly. I find this leads to a gap, where we put that ellipsis of endlessness. However, the movement instead down the interval (0,1] from 1 leads to a different but complementary and enriching result, using multiplicative inverses thus reciprocals and projecting a continuum beyond 1. That model if you will then attains to a point as we reach e near to zero in a continuum such that e^2 ~ 0, that 1/e = w, a hyperreal. This is the Robinson-type discussion. Where I further add, the counting numbers are picked up from the succession 1/1, 1/2, 1/3, 1/4 etc as we sink towards 0, in a way that naturally supports a continuum-based view. We then simply multiply through by -1 to get the negative side. It makes sense in this context to conceive of a continuum that involves ellipsis yes when we look at the “beyond the reals” aspect — reals here being in effect also the continuum that fills in the stepwise ladder of the counting numbers. That ellipsis is where the issues of interest come as we begin to look at descent from the claimed infinite past in finite stage steps. So, no there is no reason to make a middle wall of unbridgeable partition, a result that the surreals approach also allows us to see. BTW, this also shows us how the cardinality of the finite span continuum (0,1] — a proper subset — is in correspondence with the wider reals beyond 1 and it makes for a natural extension to hyperreals.
Clearly (0,1] is a continuum, and the relationship of multiplicative inverse leads to the hyperbolic function that takes reciprocals. There is no reason why we should not see in this way a continuum extending to ever higher values of reals as the number being inverted sinks ever closer to zero. It makes sense to see a zone that is all but zero such that e^2 ~ 0. The multiplicative inverses of these members of the interval, points on the restricted line from 0 to 1, have every right to be regarded as numbers, numbers beyond any finite value that we can assign. Transfinites. The more or less familiar hyperreals. With, the rehabilitated infinitesimals as the bridge-anchor to the continuum of numbers in (0,1]. (Note SEP here.) We can then conceptually explore this richer terrain accessed by a more powerful set of operations than +. Here, we can see w appearing and extending beyond in a further continuum, one that is somehow connected to the natural numbers and reals. The ellipsis appears beyond any definable finite, indicating that we have passed a traverse that successive finite stage addition cannot traverse. We can speak to the onward surreals but it is enough to say, subset. With aid of the ellipsis, we can step down from w, as the OP illustrates, and as the surreals also illustrate. Of course coming back down in finite stage steps will not bridge to the finite span from 0. With this broadened picture I think we can more clearly see my concerns. We can sensibly represent some Q as being comparable to w, and see that -Q_t would be a transfinite. We can see why stepping down in increments from a transfinite to get within finite span of 0 would be futile. Where, actually, stage succession is like pushing a plate on top of a sinking, push-down stack. Can we get from Q to 0 from the l-ward side by a stacking process? No. So now to bridge to the claimed infinite past, is this just a claim to finites beyond counting? I think that concept is incoherent. If we are in the beyond counting and bounding by successor range we are in the hyperreal transfinite domain. And that brings out the concern on oh we already load up a beginningless past that is infinite l-ward at any given R that can count down to 0, now. KF kairosfocus
KF, Until I get a clear yes/no answer to my #71, I'm not willing to continue to the more complex aspects of this discussion. It's essential that we each know what definitions the other person is using, for obvious reasons. daveS
DS, I have pointed out my concerns already. KF kairosfocus
KF, Do you therefore agree with my post #65? daveS
DS, the key issue is actuality. Surrounding the sequence of successive, cumulative, causally active stages with a conceptual bracket does not add any duration- to- now whatsoever that is not already present in the particular members. Thus, again, the key is what I spoke to two years ago as the ellipsis of endlessness [or I add: limitlessness or beginninglessness]. The infinity proper lies in the extension beyond any given finite member, not in the particular finite members themselves, by definition, such are exceeded, e.g by adding 1, 2, 3 . . . with that telling ellipsis there yet again. Let me add, further, that the very description of a member that it is finite implies that up to here cannot exhaust the membership, it continues beyond without limit if actually infinite. Where, there is no dynamic of stepwise succession that can traverse a transfinite span. KF kairosfocus
KF, Yes, in my illustration, the duration of the past is greater than any preassigned finite value however large. daveS
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