Gödel’s proof of the existence of God

_{Denyse O'Leary

May 10, 2020

Intelligent Design, Mathematics, Philosophy, theism}

Share: Facebook; Twitter; LinkedIn; Flipboard; Print; Email

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

DS, as part of core quantitative structure, which includes more. KFkairosfocus_{May 20, 2020
May
05
May
20
20
2020
01:01 PM
1
01
01
PM
PDT}

KF, That's an audacious move. :-)
And, I am not so sure we get to invent fundamental quantities that are arguably going to be present in any possible world . . . part of their power.
Haven't you already argued that the set of real numbers is present in all possible worlds?daveS_{May 20, 2020
May
05
May
20
20
2020
09:34 AM
9
09
34
AM
PDT}

DS, no, I am thinking, once we recognise ultra-small and ultra large as well as the in between as a unified structure of quantities, the number line continuum now stands as prior to sets we have used and our set choice has been relativised. Where, infinitesimals in particular have been haunting Math since classical times, not to mention once Calculus provided a unifying general approach. And, I am not so sure we get to invent fundamental quantities that are arguably going to be present in any possible world . . . part of their power. KFkairosfocus_{May 20, 2020
May
05
May
20
20
2020
09:21 AM
9
09
21
AM
PDT}

KF,
this stuff is not about to go away
Whew, that's a relief 😛 More seriously, I don't think anyone is trying to make this stuff go away, whatever that would mean. I do think you have developed a tendency to demand that models involving the real numbers also include infinitesimals and infinite numbers, even when the, *ahem*, designer of that model has stated otherwise. They're "on the table", so excluding them from the model is clearly a sign of suspect ideological commitments.daveS_{May 20, 2020
May
05
May
20
20
2020
06:38 AM
6
06
38
AM
PDT}

DS, bottomline, this stuff is not about to go away and once we have ultra small scale smooth domains the catapult concept, suitably applied, points to the transfinites in a unified space. R and extensions to multidimensional spaces, R^n, does not readily accept such. Of course, the popping up of intuitionistic logic with things fuzzy between is itself not without issues. I note some axioms definitely start with domains where h^2 --> 0 as an axiom in that context. KFkairosfocus_{May 20, 2020
May
05
May
20
20
2020
01:49 AM
1
01
49
AM
PDT}

PS: That's "classical" differential geometry I'm talking about, btw. The modern stuff is incredibly abstract.daveS_{May 19, 2020
May
05
May
19
19
2020
11:04 AM
11
11
04
AM
PDT}

KF, Yes, differential geometry is essentially calculus on arbitrary manifolds. You can use infinitesimals or not, depending on preference.daveS_{May 19, 2020
May
05
May
19
19
2020
10:39 AM
10
10
39
AM
PDT}

DS, history. Classical thermodynamics is a continuum theory, it does not depend on the reality of atoms or molecules. This is part of why it has a significant independence of stat mech, which definitely is about populations of particles.Going beyond, differential geometry -- Wiki: >>Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry [--> emphasis seems to be on in the ultra small. see here: http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf the opening is oh so ever suspiciously familiar!] The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field. >> -- is apparently its own development. KF PS: The dx^2 = 0 as defining threshold or order scale of infinitesimals etc stuff is familiar, it is some of what go to standard form addresses. And we are seeing fuzzy zones and line segments in continua etc. Some bells are a ringing.kairosfocus_{May 19, 2020
May
05
May
19
19
2020
10:33 AM
10
10
33
AM
PDT}

KF, The Clausius inequality makes sense as a statement involving "pretend" infinitesimals. No one can actually make a measurement of an infinitesimal physical quantity. bio of Urs Schreiber:
I am a permanent researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague. My degree is in theoretical physics. My work is about mathematical structures motivated from quantum field theory and string theory.
Hmm, this guy looks more mathematician than physicist You know how trustworthy they are... :-) However, point taken. The Schreiber post could be an example. I have great reservations about this statement he makes:
On the one hand the categorical logic of toposes allows to formally speak of the subset of the real line of elements that square to 0.
This subset has to be just {0}, because there are literally no other such numbers in R. In another space "like" R in some ways there could be others, sure.daveS_{May 19, 2020
May
05
May
19
19
2020
09:26 AM
9
09
26
AM
PDT}

PPS: Following up (while one ear is on the LA debate) I find https://ncatlab.org/nlab/show/synthetic+differential+geometry >>In synthetic differential geometry one formulates differential geometry axiomatically in toposes – called smooth toposes – of generalized smooth spaces by assuming the explicit existence of infinitesimal neighbourhoods of points. The main point of the axioms is to ensure that a well defined notion of infinitesimal spaces exists in the topos, whose existence concretely and usefully formalizes the wide-spread but often vague intuition about the role of infinitesimals in differential geometry. In particular, in such toposes E there exists an infinitesimal space D that behaves like the infinitesimal interval in such a way that for any space X [element of] E the tangent bundle of X, is, again as an object of the topos, just the internal hom TX := X^D (using the notation of exponential objects in the cartesian closed category E). So a tangent vector in this context is literally an infinitesimal path in X. This way, in smooth toposes it is possible to give precise well-defined meaning to many of the familiar computations – wide-spread in particular in the physics literature – that compute with supposedly “infinitesimal” quantities. Remark 1.1. As quoted by Anders Kock in his first book (p. 9), Sophus Lie – one of the founding fathers of differential geometry and, of course Lie theory – once said that he found his main theorems in Lie theory using “synthetic reasoning”, but had to write them up in non-synthetic style (see analytic versus synthetic) just due to lack of a formalized language: “The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following. I found these theories originally by synthetic considerations. But I soon realized that, as expedient ( zweckmässig ) the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which deal with objects that till now have almost exclusively been considered analytically. After long vacillations, I have decided to use a half synthetic, half analytic form. I hope my work will serve to bring justification to the synthetic method besides the analytical one.” (Sophus Lie, Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung, Math. Ann. 9 (1876).) Synthetic differential geometry provides this formalized language.>> In short, this issue is live.kairosfocus_{May 19, 2020
May
05
May
19
19
2020
09:16 AM
9
09
16
AM
PDT}

DS, actually, they are used; famously the second law of thermodynamics, in the Clausius statement giving dS GTE d'Q/T, which then feeds into the wide apparatus of Thermodynamics then combines 1st and 2nd laws which is a very powerful expression and in many other places. Yes, small increments similar to economics are also used. KF PS: Look here too https://physics.stackexchange.com/questions/92925/how-to-treat-differentials-and-infinitesimals note: >>There is an old tradition, going back all the way to Leibniz himself and carried on a lot in physics departments, to think of differentials intuitively as "infinitesimal numbers". Through the course of history, big minds have criticized Leibniz for this (for instance the otherwise great Bertrand Russell in Chapter XXXI of "A History of Western Philosophy" (1945)) as being informal and unscientific. But then something profound happened: William Lawvere, one of the most profound thinkers of the foundations of mathematics and of physics, taught the world about topos theory and in there about "synthetic differential geometry". Among other things, this is a fully rigorous mathematical context in which the old intuition of Leibniz and the intuition of plenty of naive physicists finds a full formal justification. In Synthetic differential geometry those differentials explicitly ("synthetically") exist as infinitesimal elements of the real line. A basic exposition of how this works is on the nLab at differentiation -- Exposition of differentiation via infinitesimals Notice that this is not just a big machine to produce something you already know, as some will inevitably hasten to think. On the contrary, this leads the way to the more sophisticated places of modern physics. Namely the "derived" or "higher geometric" version of synthetic differential geometry includes modern D-geometry which is at the heart for instance of modern topics such as BV-BRST formalism (see e.g. Paugam's survey) for the quantization of gauge theories, or for instance geometric Langlands correspondence, hence S-duality in string theory.>> Things are complicated.kairosfocus_{May 19, 2020
May
05
May
19
19
2020
08:49 AM
8
08
49
AM
PDT}

KF, Physicists certainly use infinitesimals informally. And for them, it's the right tool for the job. They're doing physics, not mathematics. However, I have yet to see a case where a computation (in the physical sciences, say) involving infinitesimals cannot be replaced by a computation involving only "standard" quantities (no infinitesimals, no infinities). That is, I haven't seen that *R is indispensable for the empirical sciences. Do you know of any cases where it is? Edit: This post on stackexchange is interesting. A quote from a physicist in the thread:
Actually, we never really use infinitesimal quantities. We just pretend that we do. What we mean by those is that if someone decided to make a measurement, they could find a small enough, but finite, measurement interval on which the differential relationship is "good enough". Calculus is merely the mathematical approximation for these finite differences that is highly convenient to work with. In reality everything that starts with a d in physics is really just a small enough Δ. When you see mathematicians who are trying to formalize physics, you are seeing wasted effort.
(I actually enjoy reading the results of that "wasted effort" so I don't agree that it's completely wasted, but from a physics perspective that makes sense).daveS_{May 19, 2020
May
05
May
19
19
2020
08:19 AM
8
08
19
AM
PDT}

DS, that zone *0* is not beyond N, and is around the mid point of R. KF PS: Part of my problem is precisely, the free use of *0* in the physical sciences.kairosfocus_{May 19, 2020
May
05
May
19
19
2020
07:53 AM
7
07
53
AM
PDT}

KF,
The problem would be that R would effectively become R* once transfinites are on the table and once infinitesimals were rehabilitated by what, the 1940’s.
I think that R would remain R, regardless of what developments take place externally. We don't have to tart it up by augmenting it with whatever anyone places "on the table". It's a free country, so you can do whatever you please, but I have yet to see a compelling use case for the hyperreal numbers in the empirical sciences. That's the challenge, in my view: Find a situation in the sciences where *R is indispensable. If you are just interested in the mathematics of *R, that's fine, but of course there's no end to the crazy mathematics we can dream up.daveS_{May 19, 2020
May
05
May
19
19
2020
07:48 AM
7
07
48
AM
PDT}

JVL, once hyperreal transfinites and their duals under 1/x as hyperbolic catapult are on the table, the situation changes; this has been discussed several times above, hence the discussion of fuzzy borders and neighbourhoods. In particular we have h and H as mutual images under 1/x such that h is closer to 0 than 1/z for any z in the counting numbers {0,1,2 . . .} and H is (hugely) larger than any z so delineated. We then have the cloud I for convenience designated *0* -- taking in additive inverses -- and by vector addition a similar cloud surrounds any z or even any r in R mileposted by Z. Indeed, [0,1) and (-1,0] can similarly be extended between mileposts. So, once infinitesimals such as Newton's h or Leibniz's dx are in play things are not quite the same. That problem will not go away. The hyperreals are there, the surreals beyond are there. DS, The problem would be that R would effectively become R* once transfinites are on the table and once infinitesimals were rehabilitated by what, the 1940's. I think the line of numerical quantities is what is truly foundational, and R is not an adequate model for its full span. KFkairosfocus_{May 19, 2020
May
05
May
19
19
2020
07:33 AM
7
07
33
AM
PDT}

KF (& cc to JVL)
the presence of infinitesimals implies that Reals cannot strictly bridge to 0
This wouldn't have been a problem if we had just stuck with R and hadn't bolloxed it up. :-)daveS_{May 19, 2020
May
05
May
19
19
2020
06:59 AM
6
06
59
AM
PDT}

KF,
I don’t mean forming the set basis for a field etc, but covering the range of valid numbers.
Well, I guess we can continue jamming more stuff in there until it's utterly incomprehensible 😛 But already, with the hyperreals, we have: * this weird nonconstructive ultrafilter step * different choices of ultrafilter apparently may produce different hyperreal fields (I don't know if this is completely settled) * Is there a practical way to represent hyperreal numbers for purposes of computation? With the real numbers, there is a unique non-terminating decimal (or binary, or whatever you prefer) representation that is very intuitive. It's easy to implement libraries to perform arithmetic on (certain finitely -describable) decimal expressions. I haven't found such a thing for "the" hyperreals. I've seen suggestions that this is possible using quotients of polynomial expressions, but I haven't found any hyperreal libraries for common programming languages. * Finally, that diagram at the top right of the hyperreal wikipedia page could be extended "above" and "below" indefinitely, I believe. If we consider the normal real numbers, the corresponding number line looks like: −2, −1, 0, 1, 2, multiplying each number by ε, we end up with: −2ε, −ε, 0, ε , 2ε , (or *0*) Multiplying by ε once more, we get an even more infinitesimal neighborhood of 0, perhaps **0**: − 2ε^2, − ε^2, 0, ε^2, 2ε^2, and so forth. If we multiply by ω = 1/ε repeatedly, we move up the diagram, where the "mileposts" now are in multiples of ω^n for integers n. So we have an infinite hierarchy of neighborhoods of 0 (and of every other hyperreal), going up and down. This is great fun for a mathematician, I'm sure. Is there any point for a physicist to prefer *R over R, however?daveS_{May 19, 2020
May
05
May
19
19
2020
06:00 AM
6
06
00
AM
PDT}

Kairosfocus: There are no gaps in the reals: pick any two real numbers and you can ALWAYS find another one in between them. The Reals are everywhere dense. the presence of infinitesimals implies that Reals cannot strictly bridge to 0, You can get as close as you want with the Reals. Remember: Calculus was put on firm ground with limits which do not involve infinitesimals.JVL_{May 19, 2020
May
05
May
19
19
2020
05:59 AM
5
05
59
AM
PDT}

DS, that's the fuzziness zone in action, once the ultra small is in the door given we cannot finitely bound 1/z, z in N mileposting R, as there is no definable last z_f so z_f + 1 --> w. Where not even w counts as we can conceive of w/2 etc. I begin to appreciate why the older mathematicians viewed both infinities with horror; and, continuum, thou art a real headache. They REALLY didn't have tools to handle such. KF PS: I don't mean forming the set basis for a field etc, but covering the range of valid numbers, even bound ones that are in that sense finite, exceeded by other valid numbers. And yes, the infinitesimal cloud is meant, h and relatives closer to 0 than 1/z for any finite z in N. Or 1/r for the interleaved reals. Mileposting counts.kairosfocus_{May 19, 2020
May
05
May
19
19
2020
05:14 AM
5
05
14
AM
PDT}

KF,
I am thinking, this brings in troubling questions as to just how complete the Reals are; the hyperreals look more fundamental to me, though they are challenging.
My answer: The real numbers are even more complete than the hyperreal numbers in a sense. They satisfy every definition of completeness on this page, while the hyperreal numbers do not. For example, in the real numbers, every nonempty set which is bounded above has a least upper bound or supremum (that's one form of completeness). For example, the least upper bound of (−1, 1) is 1. That's the smallest real number which is greater than or equal to every number in (−1, 1). The hyperreal number system does not satisfy this property. I think you're using the notation *0* for the set of all hyperreals which lie at most an infinitesimal distance from 0. This set does not have a least upper bound. 1, 1 − ε, 1 − 2ε, 1 − 3ε, and so on are all upper bounds for *0* but there is no smallest one.daveS_{May 19, 2020
May
05
May
19
19
2020
05:07 AM
5
05
07
AM
PDT}

F/N: I add, the hyperreals give us a way to unify numbers from infinitesimals to transfinites on a common scale with counting numbers, integers and the reals they milepost. That in turn lends teeth to the point that a transfinite span can legitimately be shown by using ellipses on a number line and that such cannot be spanned in finite stage steps. Where, too, we see that two primary fuzzy zones appear as the onward count of the naturals mileposting the reals transitions somehow into transfinites and as 1/z in N fails to get as close to 0 as h = 1/H, H a transfinite hyperreal. The price we pay is that on this extended line of quantities -- an ordered set and structure -- the presence of infinitesimals implies that Reals cannot strictly bridge to 0, one must go to infinitesimals to complete the continuum there. And as the infinitesimal cloud on 0, *0*, can then be added as a vector cloud to any r in R or any transfinite H, this similarly pervades the expanded number line. Where of course once two directions are present in the number line, we have 1-dimensional vectors: size, direction and continuously variable magnitude. I am thinking, this brings in troubling questions as to just how complete the Reals are; the hyperreals look more fundamental to me, though they are challenging. R now takes its place as finite, non infinitesimal numbers mileposted by N. All of this then feeds back into debated issues. The things that lurked in 2nd to 4th form Math have turned out to be crucial. I suspect, inchoately, we were playing with R* not R, and in College physics, it was more than playing with. Mind you I am now finding revisions on C17 - 19 Math that argue that the early greats were not as loose with infinitesimals as has been alleged. Going to standard part looks a lot like taking a limit with hyperreals, admittedly ticklish with convergence issues. How do you define a sequence of partial sums and ever tightening zones in what are already transfinite members of a sequence? Do enough infinitesimals pile up explosively, given exponentially transfinite transfinites? Etc, just feeling around half blind here. Zeno's revenge? KFkairosfocus_{May 19, 2020
May
05
May
19
19
2020
03:49 AM
3
03
49
AM
PDT}

Ds, the point is, the genie is out of the bottle, once transfinites are there and the link to infinitesimals. . KFkairosfocus_{May 18, 2020
May
05
May
18
18
2020
11:42 AM
11
11
42
AM
PDT}

KF, There's no need for them to "go away". They just aren't necessary for doing calculus is all.daveS_{May 18, 2020
May
05
May
18
18
2020
08:48 AM
8
08
48
AM
PDT}

DS, they are there in the concept space, it isn't going to go away. KFkairosfocus_{May 18, 2020
May
05
May
18
18
2020
08:38 AM
8
08
38
AM
PDT}

KF, If by "lurking", I assume you mean that we can do calculus using infinitesimals if we choose. But it's strictly optional.daveS_{May 18, 2020
May
05
May
18
18
2020
07:26 AM
7
07
26
AM
PDT}

DS, it gets more and more curious. The problem is, R is no longer alone. And, of course, infinitesimals lurk given Calculus. KFkairosfocus_{May 18, 2020
May
05
May
18
18
2020
07:07 AM
7
07
07
AM
PDT}

KF, Continuing as I have time:
The continuum number line if limited to unaltered reals is then sieve-like, with tiny fuzzy gaps essentially everywhere.
When we think of R as being embedded in *R, it is spread out, with "gaps" in a sense. But R under its usual topology has no gaps. It wasn't until we messed with R by replacing each point with an uncountable interval that these gaps appeared.daveS_{May 18, 2020
May
05
May
18
18
2020
06:31 AM
6
06
31
AM
PDT}

KF,
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.
I think I get it (partially) now. I'll reflect my understanding of the first paragraph back to confirm: Referring to the diagram on the hyperreal wikipedia page, the pink "number line" on the lower left shows that in the hyperreals, the point where 0 lies in the real number line has been replaced by what looks like an entire real number line, where the units are marked as multiples of ε. This means, for example, that (−ε, 0) and (0, ε) are intervals of positive length in *R which contain no real numbers. That is, these are "gaps" in R, if we view it sitting inside *R. **** I would caution that there are no "gaps" in the real number continuum, however, when we view it in its own standard topology. In fact, in some ways, the set of hyperreals has "holes" in it which do not exist in the reals. For example, there are Cauchy sequences in *R which do not converge.daveS_{May 18, 2020
May
05
May
18
18
2020
05:47 AM
5
05
47
AM
PDT}

DS, a nonstandard, infinitesimally altered real would be a number not reachable by a power series of rationals that exclude an infinitesimal tail . . . effectively it cannot be written as a whole + place value decimal continued to w steps. In that sense pi + dr is ontologically different from pi, it is a hyperreal in the pi-cloud *pi*. Where, as the tail dr is unique, reduction to standard form of course applies, i.e. the cloud *r* is inseparably attached to r. Hence, r is a standard real limit. This gives teeth to how dr is the catapulted dual of a transfinite. Truly, truly, truly weird but somehow wonderful, I am beginning to see why there was such an intensity in the infinitesimals debate. The smeared out real, *r* looks a lot like the figures can be made from infinitesimal line segments. Jump to the complex numbers, turn this into a 2-d cloud and voila, plane figures are patterns played with these vector-tip clouds. Multiple dimensions beckon. Do you recall, path integrals,curvy line length results and how fields come in through path integrals in physics . . . part of that uneasy trickery I spoke to. KFkairosfocus_{May 18, 2020
May
05
May
18
18
2020
12:13 AM
12
12
13
AM
PDT}

BO'H, the real issue is, can it be made continuous? See the Quora link. KFkairosfocus_{May 18, 2020
May
05
May
18
18
2020
12:02 AM
12
12
02
AM
PDT}

1 2 3 … 9 Next

You must be logged in to post a comment.

Leave a Reply