I have seen it argued that Gödel's Incompleteness Theorems have implications regarding the existence of God. Arguments for the existence of God run mostly along the lines: "Because of Gödel's Theorem, truth transcends human understanding, and therefore there is God". Arguments against God go like this: "Because of Gödel's Theorem, omniscience is impossible, hence an all-knowing God cannot exist".

Personally, I fail to see sense in such reasoning (of course, this does not necessarily say much, because I could be missing something). Given that nowadays people hold all sorts of irrational views, I can't say I am surprised --- but I would be if a serious and respectable person supported such arguments. In fact, I only saw such arguments either in the process of being rebutted, or expressed by people whom I find it very hard to take seriously.

I would be very grateful if someone could respond to these questions:

  1. Are there legitimate applications of Gödel's theorems to the existence of God, or theology in general?

  2. Do any significant philosophers or theologians ever express views of this kind?

  • 1
    This guy seems to be pretty on the ball. – Quinn Culver May 29 '13 at 22:17
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    @QuinnCulver: I remember seeing that video. It was painful. – DBK Jul 20 '13 at 0:49
  • If by "on the ball" you mean "hilariously off the wall." He is comically unfamiliar with what Gödel's theorem applies to—a specific kind of set. – ChristopherE Oct 24 '13 at 19:10
  • @DBK: This truly is painful. The speaker seems to assert we wanted to prove the axioms of geometry, and generally do mathematics fully without axioms. – Jakub Konieczny Oct 24 '13 at 21:47
  • You can find my math-logical argument at <metagovernment.org/Law_of_the_Eternal>. – TheDoctor Jul 28 '17 at 21:09
up vote 14 down vote accepted

Such arguments are indeed ... shall we say "hopeless", to be polite. For a demolition job, take a look at Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.

  • 5
    Mind giving us a review on why they are hopeless? Not much of a answer as it stands. – Neil Meyer Jul 19 '13 at 14:11
  • Because it's not clear that Gödel's theorem has any implications for knowability, much less for omniscience, much less for anything to do with gods. See answer here: philosophy.stackexchange.com/a/314/4504 – ChristopherE Oct 24 '13 at 19:16
  • Or perhaps we can say that God has access to second order logic, for which Godel falls. – Joshua May 18 '15 at 0:10

Godel's theorem says nothing about human understanding. It only places limits on certain formal axiomatic systems. Humans have ways of understanding that transcend formal axiomatic systems; for example, we can extend a given axiomatic system to prove the truths that were unprovable in the unextended system.

As an example, Zermelo-Fraenkel set theory (ZF) can neither prove nor disprove the Axiom of Choice (AC). But we can extend ZF to ZFC, and thereby prove AC.

This has nothing at all to do with theology. It's strictly a matter of formal mathematical proofs regarding axiomatic systems.

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    "But we can extend ZF to ZFC, and thereby prove AC". Since "C" in "ZFC" stands for the axiom of choice, this example is a bit weird. In ZFC, AC is added as an axiom, it is not proved anywhere. – DBK Jul 20 '13 at 12:37
  • I could have made the same point by saying that in ZF we can't prove Zorn's lemma but in ZFC we can. It comes down to the same thing. The main point being that Godel's theorem says nothing about "human understanding" as the original questioner believes. It's strictly a mathematical statement about formal axiomatic systems. – Janet Williams Jul 20 '13 at 21:34
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    I agree with your main point. Please consider that comments are (among other things) means to point out smaller errors and to suggest improvements. I pointed out that - contrary to what you write - AC is added as an axiom in ZFC, it is not proved. (I think we both agree on this really trivial point.) Zorn's lemma is indeed a better example, as its equivalence with AC can be proved in ZFC. You could improve your answer by using this example. BTW, welcome to Philosophy.SE and thanks for contributing! :) – DBK Jul 21 '13 at 0:56
  • @DBK, if you add some axiom A to your formal system, technically, proof is defined in such a way that it would be perfectly OK to say 'I can prove A'. – user132181 Mar 29 '14 at 19:55

Gödel's incompleteness theorem is based on: "The true reason for the incompleteness that is inherent in all formal systems of mathematics lies in the fact that the generation of higher and higher types can be continued into the transfinite whereas every formal system contains at most countably many.... In fact we can show that the undecidable statements presented here always become decidable by adjunction of suitable higher types .... Same holds for the axiom system of set theory. [Kurt Gödel: "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I", Monatshefte für Mathematik und Physik 38 (1931) p. 191]

Since the "generation of higher and higher types" is invalid, at least according to some branches of modern mathematics (e.g., Constructivism or MatheRealism), one cannot derive a general proof of God's existence from the incompletenes theorems.

But Gödel himself is certainly what you consider "a serious and respectable person". It is not his incompleteness theorem that proves or supports the idea of the existence of God but a direct logical proof by the greatest logician of the last century, namely by Gödel himself: Christoph Benzmüller, Bruno Woltzenlogel Paleo: "Formalization, mechanization and automation of Gödel's proof of God's existence", arXiv (2013)

The legitimate applications of Godels theorems are to mathematics, anything outside of that, especially to applications to theology is generally a form of mathematical superstition, it's a contemporary form of numerology or astrology.

There exists legitimate applications to existence of God, or theology in general, if Cantor is right and all real numbers exist, even those which humans cannot define (we can define only a countable set of real numbers). That means the existence is guaranteed by a God or a Goddess who have a list or something like that, a complete universe, for instance, of all real numbers. Yes, if moldern set theory is correct, then it proves God's existence. This will fullfil Cantor's dreams: "The time is not far, however, that my teaching will turn out to be a really exterminating weapon against all pantheism, positivism and materialism." [G. Cantor, letter to J. Hontheim (21 Dec 1893)]

  • God can't have a list of all real numbers, nothing can because such a thing cannot possibly exist. You can't use Cantor's work to try and prove the opposite of what it says. – Not_Here Jul 28 '17 at 20:51
  • As your conclusions run contrary to the careful arguments brought forward by Cantor, could you please provide context or sources for the claim that there needs to be a "list"? Most philosophies speak of God's mind containing the whole as intellectual intuition, but a list is made out of parts constituting the whole, i.e. discursive. – Philip Klöcking Jul 29 '17 at 15:01
  • @Philip Klöcking: If there are all real numnbers, then they cannot be in reach of humans because there are only countably many definitions. So where are they? Only a God can know them. How can he know them? They have to be stored somewhere. Whether you call this a one-dimensional list or a multi-dimensional memory does not matter. – Ibrahim Abd el Faruk-Shaik Jul 29 '17 at 21:53
  • @Not_Here: Of course such a list cannot possibly exist. Therefore either God can overcome this problem, or the set of real numbers is not uncountable, or it does not exist. Nowhere. I cannot see further alternatives. – Ibrahim Abd el Faruk-Shaik Jul 29 '17 at 21:57
  • Why should they be anywhere in the first place? 1.84736*10^937362638472628274737373826368282927262873628362672882626282736 can be construed. That's all that matters! It is not existing anywhere, why should it?! It's not like I just picked something out of the stock of God's thought. I randomly hit the keyboard. This is a pseudo problem. – Philip Klöcking Jul 29 '17 at 22:12

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