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This question of highest importance for everybody could not be answered in MO and will not be answered in Mathematics.SE. I can understand why the set-theorists there dive and attack it like vultures. I feel very sorry. Perhaps the prevailing understanding of logic and tolerance is here more suitable?

Background: Gödel proved the existence of God in a relatively complicated way using the positive and negative properties introduced by Leibniz and the axiomatic method ("the axiomatic method is very powerful", he said with a faint smile). To formalize the idea of a positive property, Gödel introduced a positivity operator. Just as a predicate or property provides a truth-functional assignment to individuals. We say that Pos(F) is true if F is a positive property. Please look it up if you have not yet heard of:

http://www.stats.uwaterloo.ca/~cgsmall/ontology.html
http://userpages.uni-koblenz.de/~beckert/Lehre/Seminar-LogikaufAbwegen/graf_folien.pdf

Couldn't the following simple way be more effective (and wouldn't it be appropriate to count it as belonging to mathematics)?

1) The set of real numbers is uncountable.
2) Humans can only identify countably many words.
3) Humans cannot distinguish what they cannot identify.
4) Humans cannot well-order what they cannot distinguish.
5) The real numbers can be well-ordered. (This is a theorem of set-theory: E. Zermelo: "Beweis, daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904) 514-516)
6. If this is true, then there must be a being with higher capacities than any human. (It has been proved that humans cannot well-order the real numbers. No set-theoretically definable well-ordering of the continuum can be proved to exist from the Zermelo-Fraenkel axioms together with the axiom of choice and the generalized continuum hypothesis by S. Feferman: "Some applications of the notions of forcing and generic Sets", Talk at the International Symposium on the Theory of Models, Berkeley, 1963)

QED

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    How in the world does this this contrived set of claims prove the existence of God? What do they even have to do with the idea of a divine being?
    – Cody Gray
    Commented May 1, 2012 at 17:47
  • Georg Cantor, the founder of set theory proved the existence of infinite sets by means of the holy bible. He wrote 1883 in a letter to Lipschitz: "Exodus, cap. XV, v. 18, Dominus regnabit in infinitum (aeternum) et ultra". Further evidence for his idea of aktualized or completed infinity, he took from Augustinus: S. Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui dicunt ea, quae infinita sunt, nec Dei posse scientia comprehendi. (letter of 1886).
    – I K Rus
    Commented May 3, 2012 at 10:02
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    Background information like that needs to be in the question; that was sort of the point of the comment. And Latin also needs to be translated into English. (If you don't know the translation, you probably shouldn't be quoting it, since that suggests you don't know what it means.)
    – Cody Gray
    Commented May 4, 2012 at 2:41
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    Euler’s proof: (a+bⁿ)/n = x, hence God exists.
    – J. C. Salomon
    Commented May 23, 2014 at 14:52
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    The real bummer in your argument is that you read the words of a mathematical theorem and assume a meaning that isn't there. "Every set can be well ordered" does NOT mean "for every set there is a being which can well-order the set".
    – gnasher729
    Commented Jul 22, 2015 at 6:22

4 Answers 4

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First of all: let's leave "God" out of this. The argument offers no definition of God, or why one must exist-- but that's just the least of the problems. In fact, the number of fallacies in the proposed syllogism is staggering.

There is absolutely no relevant distinction that can be drawn between "words" and "numbers" in terms of this argument. The set of real numbers is an abstract set, as is the set of potential words. Neither is countable, yet humans are capable of distinguishing between any two numbers or words, and ordering them according to numerical/alphabetical sequence. And the capability of a human to do so for larger subsets is limited only by the time allotted.

Not only does the proposed argument have no bearing on whether or not God exists, it actually fails to prove anything.

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  • Please inform yourself at least a little bit before trying to patronize. Of course the set of finite words is countable. And the set of nunbers, a subset of the former, is uncountable.
    – I K Rus
    Commented May 2, 2012 at 7:09
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    The set of finite words is countable, but the set of potential words is not-- and actual words, like actual numbers, only exist when identified. I repeat: there is no relevant distinction between words and numbers in this regard. Humans can only identify (or well-order) countably many real numbers, but the set of real numbers these instances are drawn from is uncountable-- and the same holds true for words.
    – Michael Dorfman
    Commented May 2, 2012 at 7:52
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    The set of potential words, like the set of real numbers, is an abstraction. There is no particular reason to believe that the numbers (or words) exist in any meaningful sense until they are actually used/identified.
    – Michael Dorfman
    Commented May 2, 2012 at 12:09
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    Its elements must exist? The number of questions you are begging is overwhelming. In any event, your "proof" still fails to prove anything, much less the existence of God, and you have done nothing thus far to change that. As comments are not intended for extended discussion, I suggest we move this to chat, or leave it be.
    – Michael Dorfman
    Commented May 2, 2012 at 12:50
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    Since there is an unnecessary amount of rudeness in these comments, I'm sorry I got involved. I had an idea that I shared with decent intentions. @IKRus, maybe you are right in that a first semester student can show me this error but I doubt you could find any silly enough to make the argument you did in attempting to prove the existence of god from mathematics. Thank you for the pleasant conversation.
    – Alborz Yarahmadi
    Commented May 3, 2012 at 20:34
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Like the ontological argument and most arguments of this (a priori) nature, no matter how profound the logic is, the whole argument fails to have much real-world applicability.

First I would just like to point out that instead of ending your argument with "QED", it would be appropriate to end with an actual conclusion, as any conclusion which may appear obvious to you may not be as obvious to everyone else.

I'm no mathematician, but it seems the flaw with your argument lies with premise 5; I don't see how it is intrinsically true that the real numbers can be well-ordered. Real numbers can only be well-ordered insofar as we possess the ability to identify them. That is, if you admit the set of real numbers cannot be counted (premise 1), then you must admit that the set of real numbers cannot be well-ordered.

But let's just say this argument was without error, that it was through and through quite sound. The next problem is that the reasoning you offer doesn't get us anywhere; it doesn't imply God must exist at all. Period. At best it would imply "a thing" did "something" to make this math work; that thing needn't be a God and wouldn't even necessarily exist today.

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  • You are quite right when you say: "Real numbers can only be well-ordered insofar as we possess the ability to identify them". But modern mathematics is proud to have proved the contrary (Zermelo, 1904). Hence, if modern mathematics is not wrong, then there must be a "something" that can do the well-ordering. I call it God. How would you call it?
    – I K Rus
    Commented May 1, 2012 at 20:53
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    @IKRus I would call it the axiom of choice.
    – Thomas Klimpel
    Commented May 1, 2012 at 21:04
  • The axiom does nothing. Either the axiom lies or the mathematicians who "proved": every set can be well-ordered.
    – I K Rus
    Commented May 2, 2012 at 6:47
  • How would we even have a real number yet not be able to identify it? I think you're accepting part of the argument that isn't logically sound, namely that there is actually something around that needs identification. This is true for a backgammon set, but not hypothetical members of uncountably infinite sets.
    – Rex Kerr
    Commented May 2, 2012 at 15:45
  • We cannot have a real number that is not identified - at least as the solution of an equation like "2x + 3 = e". But set theory is based upon the belief that there are more real numbers than can be identified by finite strings of symbols like the one above.
    – I K Rus
    Commented May 3, 2012 at 10:17
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You're playing word-games with "identify" and "distinguish".

Humans can only identify countably many words.

In one sense this is true, because if you ask me to identify a particular word, I need a finite representation of it. But in another sense we can't even identify countably many words; I have finite lifetime and memory and so on, so it seems implausible that I could even check whether two 1080 character long words were the same. And in another sense we can identify uncountably many words by, for example, rules that tell us whether or not a word belongs to some set. The words then need not be finite in length or in a countable set; we can't necessarily manually check their properties or identity, but we can manipulate finite sets of rules that describe them.

So already by (2) you're mired in ambiguity, and this sinks the argument when you get to (4) and (5), where in (4) you use a limited sense of identify and in (5) you implicitly use an expansive one.

Incidentally, point (3) is either false or adds absolutely nothing: if we encrypt each of our user names, I'll be able to tell that they are different (distinguish) without being able to tell which is which (identify). Or if by "distinguish" you mean "check the equivalence relation of", that's exactly what you have to do to identify something. So you're not even using a different concept here. "Humans cannot well-order items whose equivalence relation they cannot check." (This is true, but you still have the check-by-hand vs. check-using-logic dichotomy that sunk point (2).)

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  • You say: "we can't even identify countably many words". That is true, but you can identify each of them. "And in another sense we can identify uncountably many words by, for example, rules that tell us whether or not a word belongs to some set." That is not true. In order to decide whether a word belongs to some set, this very word must be given. Therefore it must be encoded by a sequence of symbols that can be communicated in finite time.
    – I K Rus
    Commented May 2, 2012 at 20:04
  • @IKRus - You can use a description that is different than the definition in the set. For example, "the ratio of the circumference to the diameter of a circle" is not infinitely long, even though the decimal representation of that number is infinitely long.
    – Rex Kerr
    Commented May 2, 2012 at 20:23
  • In fact there are many, many definitions for pi, one of them you have quoted. Other involve the definition of infinite series (Gregory-Leibniz, Euler) infinite products (Vieta, Wallis) sequnces and modular identities. In fact there are infinitely many words, each one defining pi. But there are many more real numbers without any finite definition.
    – I K Rus
    Commented May 3, 2012 at 10:22
  • @IKRus - Indeed. But who cares? You can still do mathematics. If you mean that for real numbers to actually physically exist (not just as a logical construct that we've made), we'd need something different than our known physical reality, yes, of course. But there's no evidence they do exist, so that doesn't get you anywhere either.
    – Rex Kerr
    Commented May 3, 2012 at 13:55
  • Of course we can do mathematics like before Cantor. But if there are uncountably many reals that can be well-ordered (not in the real universe - that is impossible, even if the universe is an infinite multiverse), then only God can do so. Who else? No inhabitant of the infinite universe accomplish that. Modern set theory proves both statements (look into the literature given in my question). Therefore it either proves a contradiction or it proves God (or a comparable being).
    – I K Rus
    Commented May 3, 2012 at 16:05
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The answer is no. Mathematics can't prove the existence of anything.

How math corresponds to reality is a question for the philosophy of mathematics. There are many theories. Some believe that numbers exists and others believe they don't exist. Some believe in mathematical truth while others believe mathematical truth is fictional. Some believe that math exists in minds, others believe it exists in a Platonic realm. A lot of philosophers and mathematicians simply don't care!

Mathematical infinite order might exist but before you call that "God" maybe you should ask the theologians about it?

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  • In any case, if numbers can be well-ordered, they must exist. I have also asked in Christianity.SE. But the mathematics there is on low level.
    – I K Rus
    Commented May 2, 2012 at 20:06
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    If Superman can be stronger than batman he must exist, right? We all know that Superman is stronger than batman.
    – offroff
    Commented May 2, 2012 at 20:48
  • Of course superman exists, namely as an idea that can be put in some order with respect to batman - just like every real number.
    – I K Rus
    Commented May 3, 2012 at 10:27
  • Now I'll go teach my children that batman is stronger than superman. Ideas are subjective. Gödel believed in objective math, independent of minds.
    – offroff
    Commented May 3, 2012 at 15:05
  • Ideas are subjective, poems, pictures, inventions. Numbers are ideas too, but they have a more objective status. If you teach your children 10 < 3, they will get in trouble sometime.
    – I K Rus
    Commented May 3, 2012 at 16:08

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