Gödel’s proof of the existence of God

_{Denyse O'Leary

May 10, 2020

Intelligent Design, Mathematics, Philosophy, theism}

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You didn’t know, possibly, that when he thought we was dying, he showed the notebook to one of his colleagues, who copied out the proof:

In an unsanitized, politically incorrect (but factual) history, Selmer Bringsjord talks about how the tormented genius Kurt Gödel took up a quest that dated back a thousand years to prove the existence of God by formal logic. His original version didn’t quite work but his editor’s version passed an important logic test:
“When we go to Gödel, we skip over the modern advocates of this argument. It’s harsh—I’m just going to say it—from the standpoint of someone who’s reasonably well-versed in formal logic, I think it’s a bit of a doldrums, despite some of the attention, until Gödel does his thing.
Gödel does it formally and then some folks in Germany, doing automated reasoning, verified it a few years back. They verified the version that Dana Scott copied out of the notebook. That is, what they verify is that there is no doubt; it’s machine-verified proof. So now we’re left with just the truth of the premises and how we judge them.”
News, “Gödel and God: A surprising history” at Mind Matters News

Further reading:

Faith is the most fundamental of the mathematical tools: An early twentieth century clash of giants showed that even mathematics depends on some unprovable assumptions. (Daniel Andrés Díaz-Pachón)

and

God’s existence is proven by science. Arguments for God’s existence can be demonstrated by the ordinary method of scientific inference. (Michael Egnor)

Comments

PPS: This at Quora gets wild, bringing in surreals perspectives: https://www.quora.com/Does-constructing-the-hyperreal-number-system-require-rejecting-the-axiom-of-completeness-If-not-why?share=1kairosfocus_{May 18, 2020
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Sorry, folks. No matter how hard you try, you won't be able to make this thread transfinitely long.Bob O'H_{May 17, 2020
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DS, do you not see that once the hyperbolic catapult 1/x is present, and once the infinitesimal h is admitted, using notation I think goes back to Newton, then surrounding 0 we have a cloud of infinitesimal values, *0* which are explicitly not reals but are somehow on the continuum of numbers in the interval [0,1) on the RHS part and (-1,0] on the L-ward side? If there are numbers legitimately in the continuum but that are NOT reals, is there not, then, a fuzzy gap in the reals as asterisked? As in the v close neighbourhood of 0. As this is a zone, and as the continuum number line is a vector space, we can shift it R/L-ward by tip to tail and infinitesimally alter any given r in a similar way to 0, just "add" the members of the cloud to r. That means R is pervaded, riddled everywhere with similar gaps, once the primary one is established. The continuum number line if limited to unaltered reals is then sieve-like, with tiny fuzzy gaps essentially everywhere. Or else, we accept the smearing out of reals, r --> r + *0* --> *r*, and so the continuum as a whole is hyperreal, if it is to be "truly" continuous. This already shows the notion of a continuous number line is in a sense independent of the set, R [taken as excluding infinitesimals and other numbers mileposted by Z but infinitesimally altered, i.e. r + dr is here not a real, allowing reduction to "standard" form]. Once it is so taken, the truly continuous number line is hyperreal. Not, real. And if we insist R is truly continuous, then it looks like infinitesimals and infinitesimally altered numbers would have to somehow be brought in. Nonstandard reals, I suppose. Then, we see 1/h --> H, unified with the real span mileposted by Z. The same tip to tail smearing can be applied to H, so we have *H* too. This means the continuum goes to transfinite realms and we have infinitesimally altered transfinites. H + dH would be a valid part of the transfinite span, just that H is transfinitely remote from 0 and 1/H --> h. Do we dare say, nonstandard transfinites? I dunno, this is following the logic as it comes up. I find myself stunned by the idea of doing a Calculus move in transfinite realms. Coming back through the catapult, if we insist R is truly continuous then infinitesimals are locked in it seems, in which case given a unified gamut of numbers, transfinites knock at the door. So, are the reals truly the continuum mileposted by Z? Where do h, w, H and co fit in? [It is 150 years too late to lock them out.] Weird. The fuzzy zones seem to have teeth. KF PS: a related discussion https://boxingpythagoras.com/2017/09/30/infinitesimal-calculus-1-the-numbers-between-numbers/ note too the wheel paradox illustration pointing to a continuum cardinality.kairosfocus_{May 17, 2020
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😳 I have to admit I can't make head or tail out of the last part of #200 or #208.daveS_{May 17, 2020
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DS, the principal fuzzy zones are there, such as -1 ----*0* ---- 1 and that as the finite values traverse to transfinite ones, we can represent (* . . . -0-1-2- . . . *). The one at: *0* can then be vector displaced all along the line, showing the interwoven nature of R and R*. As well of course: (r-1--*r*---r+1) for any r in R. However, I had pointed to the continuum as we go v close to 0. With hyperreals not in view, we would not see it but there is an issue once infinitesimals of order h are in view. Where the hyperreals are not about to go quietly into the good, good night. So, it looks like we really do need to ponder the interwoven nature of hyperreals and reals, with every real subject to infinitesimal augmentation by displacing a copy of *0* as, r + *0*, Things have got rather hairy, on reflection and R* seems in some ways more fundamental and comprehensive. KFkairosfocus_{May 17, 2020
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KF,
notice the issue on the continuum vs the gap next to 0? A fuzzy gap?
Er, no? Not that I'm aware of. Perhaps your #206 is a clarification or correction? Anyway, it would seem we can dispense with expressions such as "fuzzy zone".daveS_{May 17, 2020
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DS, yes, by secondary extension, a cloud of infinitesimally altered numbers clusters around each real, which fits with Calculus. The point is, the hyperreals are pervasive once they are in the door. However, the zone to 0 and that as the naturals-mileposted reals run away to the transfinite are primary. My sense of what the continuum is, is thus affected, and of course any material zone of a continuum gas continuum cardinality, with the reals line giving a definite ordering. KFkairosfocus_{May 17, 2020
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DaveS: In the real numbers, any two nonempty open intervals (a, b) and (c, d) can be put into one-to-one correspondence via an order preserving function. The two intervals “look” exactly the same except for the labels on the numbers. I was going to say: Good luck with that! But then I figured out a way to do if for any open intervals. I think. Anyway, carry on.JVL_{May 17, 2020
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DS, notice the issue on the continuum vs the gap next to 0? A fuzzy gap? And yes, the continuum has an order, howbeit, that order does present the fuzzy zones whenever finite to transfinite or finite to infinitesimal transitions occur. And, it is too late to get rid of those. KFkairosfocus_{May 17, 2020
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KF, The issue I see is that all "zones" in the hyperreals are equally fuzzy, as I understand the meaning of that word. In the real numbers, any two nonempty open intervals (a, b) and (c, d) can be put into one-to-one correspondence via an order preserving function. The two intervals "look" exactly the same except for the labels on the numbers. I believe* the same holds in the hyperreal setting. Imagine standing x1 = 0 and at x2 = 1/ε. Then the neighborhoods around these two points look exactly the same, in a way that can be made precise (I think). *I don't have a reference for this, but will look for one later.daveS_{May 17, 2020
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DS, a key one is just next to 0, of course, the real home of the infinitesimals. A second one will be between some arbitrarily high but specific n in N and w/2 [etc] as is shown in the use of the ellipsis, another will lie between w/2 and w as we cannot count up in steps from w/2 to w or down in steps from w to w/2 and there are many, many more. Indeed, I am reminded of the point that continuum on [0,1) raises the interesting point that h is in the continuum but is not regarded as a real. So, do we hold that there is a forbidden zone and hidden tiny gap . . . one as dense with points in R* as the whole transfinite realm . . . just away from 0 where the infinitesimals would fit, or do we reckon instead that the hyperreals allow a better model of the continuum, one well fitted to say Calculus? Or do we banish the infinitesimals and their hyperbolic catapult 1/x duals the transfinites, holding that there is no valid number beyond what n in N can milepost? The more I look at it the more central R* seems to be, and the more significant those odd puzzles in 2nd to 4th form were. KFkairosfocus_{May 17, 2020
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Edit: I see the update now. There is a diagram on the wikipedia page here at the top right. I can't directly link to it. Can you describe where these zones lie on it?daveS_{May 17, 2020
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DS, I already pointed out the structural feature that the zone TO the recognised transfinites is fuzzy (reflected in the significance of the ellipses and the n and onward problem where you cannot exhaust to some n_f so n_f + 1 --> w), similar to the fuzzy zone to the infinitesimals and even the open border side of an interval in a continuum such as [0,1). I made a comparison to the old Roman borders where a Wall was built on defensible ground and forays and influence beyond faded out as one went beyond a zone of feasible control -- in some ways it is a historical pity they could not sustainably move beyond the Rhine to a shorter river line, but that is a perennial challenge of continental Europe; the fall of the W Roman Empire cost us centuries of development. Back on subject, I am of course bearing in mind the related but not identical idea of fuzzily bordered, partial membership sets. Given part-ways to a full integration or differentiation operation, even operators it seems can have fuzzy borders. So, the ellipsis is material. What we can explicitly count to or represent is finite but always with an elliptically loaded beyond. Surge forth here and the Barbarians retreat beyond reach, then when you pull back, they return. KF PS: Where h is a value LT 1/n for any specific n in N, such that in the interval (h,1] there is no definable closest neighbour to h, and yet there are valid numbers everywhere in [0,1], including in [h,1], then we see that by catapult, beyond any specific n in N, there is a valid CONTINUUM with milestones that cannot be counted exhaustively all the way to H = 1/h, and particularly we may identify on w as order type of {0,1,2 . . .} that there is no identifable n in N, n_f so that n_f + 1 --> w. Thus, w is a limit ordinal on N, and this points to transfinite induction etc. So, while not a precising definition, we can characterise the fuzzy zones by this sort of process.kairosfocus_{May 17, 2020
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KF, Would you be able to give a precise definition of this "fuzzy zone" in the hyperreals?daveS_{May 17, 2020
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PS: Notice on the general issues of cosmology https://uncommondescent.com/intelligent-design/at-astronomy-mag-is-the-big-bang-in-crisis-rob-sheldon-responds/ I note, the HR plots of star clusters with breakaways from the main sequence heading to the giants bands and relative absence of White Dwarfs point to the same general t-frame as expansion, so that ~ 14 BY is a plausible, observationally anchored age for a clearly expanding cosmos. Beyond that, the incrementally built models over the past 50 years seem to be in trouble.kairosfocus_{May 17, 2020
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F/N2: Associated, I note on the quantitative structure of R*, including R in context. We start with countable numbers mileposting R, where between any n and n+1 we can wedge in a vectorially displaced [0,1), anchored on n+0 = n; recognising the continuum. We get Z by the additive inverse, n' + n = 0, identity element for +. Here, there is no definable last positive integer n_final such that n_final + 1 --> w, first transfinite ordinal. There is a fuzzy zone as n goes ever higher, where every n we can represent or explicitly count to is bound onward by n+1, n+2 etc. So we can match that to 0,1,2 etc. This is related to how we recognise the transfinite nature and it is why the ellipses in Z := { . . . -2, -1, 0 1, 2 . . . } do a lot of work, too often not sufficiently recognised. Also, incrementation fails to attain the transfinites, we exploit continuum in [0,1) and the hyperbolic catapult 1/x to recognise an infinitesimal h where h is less than 1/n for any n in N, and in turn 1/h gives a hyperinteger H greater than any n in N. (We here have chosen h strategically.) This wider span then opens up similar extensions and additive inverses, yielding the transfinite spans of the hyperreals. Infinitesimals of various character may readily be constructed algebraically on h etc. We can also see that the infinitesimal cloud around 0 can be exended to any r in R by addition, and can similarly cloud around H etc. We have a unified number space, extensible to the complex domain etc. I think this domain is useful in itself, just as a unified structure of quantity. It also obviously extends to the Calculus. Indeed, I have seen the suggestion that reducing an infinitesimally augmented result to standard form per Robinson is closely comparable to the limit on the now more conventional approach. My comment here is it may help in forming core calculus concepts which extend to standardised results. It is in this context that I pointed out that finite means just that and that the transfinite span cannot be traversed in steps. Even, when it is implicit and fuzzy. KFkairosfocus_{May 17, 2020
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F/N: In a sense we are full circle. The Godel argument pivots on logic of being and on the need for adequate, necessary being world root. It is somewhat abstracted from issues such as, rooting of responsible, rational freedom and associated inescapable, built in moral government. It is known to be of valid formal structure (at least in the adjusted version), and along with other similar arguments on this and other subjects, it is premises that what have to be challenged, to reject conclusions. Thus, we see the real issues: P => Q, ~Q so ~P carries with it issues of what alternative premises and what are their difficulties. Also, what does ~Q itself entail. So, we are back to issues of worldviews choice, comparative difficulties analysis and what conclusions are we willing and able to live with? Why? KF PS: Such issues become cumulatively challenging for evolutionary materialistic scientism and fellow travellers. For example, it is manifestly inescapable that our rationality is governed by duties to truth, to right reason, to prudence, to sound conscience, to neighbour (esp. fairness and justice) etc. We are under built-in moral government and also need to credibly account for rationality and the ability to warrant sound knowledge. In that context, immediately, any worldview that undermines rational, responsible, morally governed freedom is in the end futile. Where, reduction of rationality to computing on a GIGO-driven substrate confuses ground and consequent, freely taken inference governed by first duties of reason, with blindly mechanical and/or stochastic, cause-effect processes driven by the organisation of a substrate is patently self-referentially incoherent. That already decisively undermines evolutionary materialistic scientism and its fellow travellers. Taken another way, such views try to ground mind in what is grossly inadequate, computation. Similarly, the inescapable first duties confront us with the challenge of freedom: a free creature is necessarily, free. Thus the is and the ought diverge and choices (sometimes thorny ones) must be made. However the IS-OUGHT gap now confronts us, which, post Godel, can only be resolved and bridged at the reality root. Such then turns into the issue of roots of a causal-temporal order. Utter non-being is a non-starter. Circular causation where a not yet future reaches back retrocausally is much the same [and yes, there are those who suggest and imply such]. This means there are two serious options, the root is an infinite regress of the material, causal-temporal order, or else there is a finitely remote completely causally adequate reality root that is non-physical as a necessary being. A subsidiary claim is that necessary being is a dubious or arbitrary construct. I trust the cross-world reality and consequent rational power as well as applicability of numbers suffices to show otherwise. Somewhere, there is a reality root causally independent of external cause. Much of the thread then turned into a retracing of some aspects of the exchanges over several years. Plainly, an explicitly transfinite span cannot be traversed in finite stage steps, more or less almost by definition of a countable transfinite. The real issue is, are we dealing with an implicitly un-spannable transfinite that cannot be exhausted by similar steps? I suggest that the fact that beyond any n in Z, there are n+1, n+2 etc equally matched to 0,1,2 etc carries that weight. Similarly, that if between any past stage p we can traverse from to now q, the duration d = q - p will be finite, so if all traverses are finite, we have a finite span. If the body of traverses is collectively transfinite, at least one past point K' will be specifically transfinitely remote. (The claim that Z in R has in it an endless onward supply of finite values and that defines its infinitude, to my mind only shows that we have the sort of fuzzy zone I described earlier.) Similarly, to avoid issues of cumulative entropy and finite past [and future] oscillations, rather speculative and patently fine tuned . . . thus credibly contingent . . . observationally unconstrained models of cosmological origins have had to be constructed. For instance, Wiki summarises:
The Steinhardt–Turok model In this cyclic model, two parallel orbifold planes or M-branes collide periodically in a higher-dimensional space.[8] The visible four-dimensional universe lies on one of these branes. The collisions correspond to a reversal from contraction to expansion, or a Big Crunch followed immediately by a Big Bang. The matter and radiation we see today were generated during the most recent collision in a pattern dictated by quantum fluctuations created before the branes. After billions of years the universe reached the state we observe today; after additional billions of years it will ultimately begin to contract again. Dark energy corresponds to a force between the branes, and serves the crucial role of solving the monopole, horizon, and flatness problems. Moreover, the cycles can continue indefinitely into the past and the future, and the solution is an attractor, so it can provide a complete history of the universe. As Richard C. Tolman showed, the earlier cyclic model failed because the universe would undergo inevitable thermodynamic heat death.[1] However, the newer cyclic model evades this by having a net expansion each cycle, preventing entropy from building up. [--> project a net expansion back and where do we go? unless, already actually infinite in expanse, on which contraction arguably evaporates?] However, there remain major open issues in the model. Foremost among them is that colliding branes are not understood by string theorists, and nobody knows if the scale invariant spectrum will be destroyed by the big crunch. [-> notice, speculative nature] Moreover, as with cosmic inflation, while the general character of the forces (in the ekpyrotic scenario, a force between branes) required to create the vacuum fluctuations is known, there is no candidate from particle physics.[9]
What is clearly happening is that the ~P and ~Q are beginning to bite. There is no credible account for the morally governed mind to come up with such speculations, indeed, rationality and responsibility would reduce to grand delusion. The commitment to a frame in which infinite causal-temporal regress lurks in the suggested implicit infinitude of numbers [where every one we can count to or represent is finite and bounded] opens a further can of worms. The associated neo-oscillating models for cosmology are openly speculative, at best. This is sounding like the ever growing patchwork that led to the collapse of the Ptolemaic cosmology 400 years ago. Faith, not founded on essentially indisputable facts of science, and beginning to be a degenerative ideological programme. Where of course, such speculative philosophy with mathematical apparatus done while wearing lab coats is not empirically constrained science. It therefore faces the full force of comparative difficulties, and it simply will not do to a priori rule out the possibility of purpose as "unscientific," as we saw with Monod. Then, the logic of necessary being speaks: a serious candidate necessary being is either impossible in the way a square circle is, or else it is actual. This is the heart of modal ontological reasoning. So, evolutionary materialistic scientism and fellow travellers either need to show God not a serious candidate or else impossible of being. On the latter, the usual argument of 50 years past is undermined, alleged contradiction in the face of evil. On the former, the bill of goods for reality root for a world with morally governed creatures should give pause even if one is dismissive to the sheer raw fact of theism as a major world view position. (Where, if so many are written off as delusional, it undermines the general credibility of the mind.) So, we have some pretty serious choices to make regarding worldview first plausibles regarding the root of reality.kairosfocus_{May 17, 2020
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EG (JVL): Actually, when there is a me too pattern without substantial cogent contribution, especially in support of an argument that may itself lack cogency, piling on becomes a question. KFkairosfocus_{May 17, 2020
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DS, In the context of physics, my concern is to moderate with the lessons learned by Abraham Robinson and others, also to put into the quiver an understanding of the full range of available quantities and structures. Then, it wouldn't hurt, to focus a bit on logic of being roots on why Logic of structure and quantity -- Math -- is so useful (per Wigner), perhaps even including a bit of possible worlds thinking, setting up a more conscious approach to relevant modelling. All of which said, it is interesting that one result of the above is a drawing out, more and more, of the sort of worldview level commitments that are incrementally required to take up and keep evolutionary materialistic scientism and/or fellow traveller views. I am very aware that P => Q can always be responded to, ~Q so, not P, but then at worldviews level, there are not that many root-level options and the cumulative commitments are sounding to me a lot like Cycles and epicycles etc c 1500 - 1650. KF PS JB's textbook is high school level, in effect; I guess in the US system, Advanced Placement. In recent years I ran across a UK GCSE Math textbook. Basic Calculus is now in what would have been ordinary not additional math in my day. Many years back, I learned that in Russia, every High School kid had to do 4 - 5 years of Calculus level Math and Physics, deemed necessities for the common defence.kairosfocus_{May 16, 2020
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KF, You can certainly do calculus with infinitesimals if you like. If it works for the physicists, that's great. For undergraduate math students, I don't think it should be a high priority. Maybe for the more advanced students, who can understand the construction.daveS_{May 16, 2020
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PS: Some reading https://arxiv.org/pdf/1205.0174.pdfkairosfocus_{May 16, 2020
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DS, I never claimed they were locked out of physics; it should be clear from above that from 2nd to 4th form High School, issues tied to R* were involved in Math-related issues I encountered, and all the more as Calculus came in. Since you brought it up, they were deeply embedded in College Physics, but there was a problem of loose C18 - 19 style use without insight from the [then fairly new, admittedly] C20 developments. I am also simply interested in opening up the number space as early as reasonable, much along JB's lines. Where there is an issue is as came out on your intervention above. KFkairosfocus_{May 16, 2020
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KF,
Hyperreals are all over the place in Physics and related fields where infinitesimals — oops, differentials — also get used as algebraic fractions.
Ok, I must have misunderstood. The hyperreal numbers are indeed not being locked out.daveS_{May 16, 2020
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KF
JVL, attempted piling on fails. KF
Translation: ‘Two or more people disagree with me.’Ed George_{May 16, 2020
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Kairosfocus: JVL, attempted piling on fails. I have no idea what you are talking about.JVL_{May 16, 2020
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JVL, attempted piling on fails. KFkairosfocus_{May 16, 2020
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DS, there is a clearly ideological domination in our civilisation, evolutionary materialistic scientism. For instance, the Nobel Prize winner Monod:
[T]he scientific attitude implies what I call the postulate of objectivity—that is to say, the fundamental postulate that there is no plan, that there is no intention in the universe. Now, this is basically incompatible with virtually all the religious or metaphysical systems whatever, all of which try to show that there is some sort of harmony between man and the universe and that man is a product—predictable if not indispensable—of the evolution of the universe.— Jacques Monod [Quoted in John C. Hess, 'French Nobel Biologist Says World Based On Chance', New York Times (15 Mar 1971), p. 6. Cited in Herbert Marcuse, Counter-Revolution and Revolt (1972), p. 66.]
That carries over into Physics as a matter of course, but that is so just about everywhere. More on point, I have suggested on experience and observation, that the hyperreals are more what we deal with, what with infinitesimals. Which BTW tend to be used rather freely in Physics as a holdover from C18 - 19 early. I used to actually mutter about mathematical trickery as a somewhat hybrid student, partly in that context. >>Here are my hypotheses: (1) Physicists/cosmologists/etc. simply find it unnecessary to use the very weird and complicated hyperreal number system, when the relatively tame set of real numbers suffices>> I already mentioned the rather free use of infinitesimals, even in improper differentials such as in d'q/T in the second law of thermodynamics. I used a dash for the actual bar. Hyperreals are all over the place in Physics and related fields where infinitesimals -- oops, differentials -- also get used as algebraic fractions. >> reformulating physical theories in terms of the hyperreal numbers would be a heckuva lot of work, where there’s no indication anything is to be gained. >> The hyperreals are already there as infinitesimals and infinitesimally altered quantities, copiously there. What would be needed is a bit of justification and advice on not getting into C18 type troubles. Your stack exchange quote may relate to renormalisation etc. My concerns start with the free use of infinitesimals, as noted. So, this tangent fails. KFkairosfocus_{May 16, 2020
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Kairosfocus: What I have done is to show that there is a serious problem, that evolutionary materialism has serious implications in terms of ultimate origins, that an assumed infinite origin is not a default, and that there are serious issues about the transfinite, that are tied to ever so many routine things I've said, several times, that I am NOT trying to push forward the ideas of a possible infinite past; it is not even my belief, It's not an issue with evolutionary issues. Why do you keep trying to make it so? For cause, what I find unsound is the idea that in a causal-temporal cosmos, with cumulative final stages to now there is an idea that we have an infinite past with durations to now from every past stage being finite. This directly implies global finitude. Instead, I find that at each occasion we specify a particular p that is finitely removed in steps from now, the onward fuzzy zone is where the transfiniteness resides. Further, duration is between specific times and if the finitude of the past remote point in question — it matters not if there are yet further onward stages — is relaxed, the gap cannot be traversed in finite stage steps. Exactly right, the gap cannot be traversed in finite steps. And any given point in the past is a finite number of steps away from the present. That leads me to conclude, there is no warrant for claiming a transfinite actual past accumulating stepwise to now. And I didn't say there was!! Especially, if one also claims that all past points are only finitely remote. I suspect, the implicitness of the fuzzy zone commonly represented by an ellipsis, is lurking unacknowledged, with force much as I outlined I'm suggesting we just leave it there and stop beating the horse past death. Really, it's time to quit. I am not going to agree with you no matter how often your repeat the same thing. You are not going to agree with me. So let's stop. Here are my hypotheses: (1) Physicists/cosmologists/etc. simply find it unnecessary to use the very weird and complicated hyperreal number system, when the relatively tame set of real numbers suffices. Or, (2) that reformulating physical theories in terms of the hyperreal numbers would be a heckuva lot of work, where there’s no indication anything is to be gained. Or, maybe, you don't grasp the mathematics as well as you think you do. Just let it go. It doesn't even matter that much. I've agreed to consider that our measurable time starts about 13.5 - 14 billion years ago. That is the materialist viewpoint. Why do you keep going on and on about it? I do not understand.JVL_{May 16, 2020
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KF, So now there is a cabal of """elites""" locking the hyperreal numbers out of physics? Here are my hypotheses: (1) Physicists/cosmologists/etc. simply find it unnecessary to use the very weird and complicated hyperreal number system, when the relatively tame set of real numbers suffices. Or, (2) that reformulating physical theories in terms of the hyperreal numbers would be a heckuva lot of work, where there's no indication anything is to be gained. Here's a paragraph from a post on the physics stackexchange concerning a related issue. We all know random posts on stackexchange are always true of course:
The question whether surreal or hyperreal numbers (that both contain the reals, even if they have the same cardinality) could be useful to provide a more satisfactory theory of QM is maybe more interesting. The mathematical evidences, such as the transfer principle for hyperreal numbers, suggest that probably a QM theory with hyperreal/surreal numbers would have essentially the same predictive power than standard QM as it is formulated, but would probably be more involved, and would have to be developed from scratch.
daveS_{May 16, 2020
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JVL, both are used. I used it the other way in my paper. I do not actually expect you to agree . . . the polarisation is too predictable for that. What I have done is to show that there is a serious problem, that evolutionary materialism has serious implications in terms of ultimate origins, that an assumed infinite origin is not a default, and that there are serious issues about the transfinite, that are tied to ever so many routine things. For cause, what I find unsound is the idea that in a causal-temporal cosmos, with cumulative final stages to now there is an idea that we have an infinite past with durations to now from every past stage being finite. This directly implies global finitude. Instead, I find that at each occasion we specify a particular p that is finitely removed in steps from now, the onward fuzzy zone is where the transfiniteness resides. Further, duration is between specific times and if the finitude of the past remote point in question -- it matters not if there are yet further onward stages -- is relaxed, the gap cannot be traversed in finite stage steps. That leads me to conclude, there is no warrant for claiming a transfinite actual past accumulating stepwise to now. Especially, if one also claims that all past points are only finitely remote. I suspect, the implicitness of the fuzzy zone commonly represented by an ellipsis, is lurking unacknowledged, with force much as I outlined. KFkairosfocus_{May 16, 2020
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