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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?

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    This guy seems to be pretty on the ball.
    – Quinn Culver
    May 29, 2013 at 22:17
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    @QuinnCulver: I remember seeing that video. It was painful.
    – DBK
    Jul 20, 2013 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, 2013 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, 2013 at 21:47
  • You can find my math-logical argument at <metagovernment.org/Law_of_the_Eternal>.
    – Marxos
    Jul 28, 2017 at 21:09

6 Answers 6

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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.

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    Mind giving us a review on why they are hopeless? Not much of a answer as it stands.
    – Neil Meyer
    Jul 19, 2013 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, 2013 at 19:16
  • Or perhaps we can say that God has access to second order logic, for which Godel falls.
    – Joshua
    May 18, 2015 at 0:10
  • @ChristopherE While the legitimacy of the applications of that theorem may be arguable, it does have implications about knowability. Namely, he did show that there are statements in Number Theory, that although true can not be proven.
    – Santiago Estupiñán
    Apr 27, 2022 at 19:15
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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, 2013 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, 2013 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, 2013 at 0:56
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    @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, 2014 at 19:55
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    +1 - But perhaps the theorem shows we cannot build a tower of logic all the way to Heaven, in which case score one for the Old Testament. .
    – user20253
    Oct 19, 2019 at 11:34
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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.

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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)

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Godel’s incompleteness theorem basically says that reason is limited.

Quantum theory seems to say that empiricism is limited.

Those limits provide gaps that open up the doors to metaphysics

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Gödel himself asserted that his result applied to theology: he submitted that there exists a Transcendent Truth of the divine kind. Being the greatest logician ever, and probably the greatest intellect ever, it’s best not to resort to “authorities” such as Torkel Franzén or any other “debunker of all things religious”.

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    Hello, and welcome to Philosophy.SE. Making claims about someone doing or writing something or even giving citations without proper referencing does not exactly strengthen your answer. Please consider adding sources for your claims. Having said that, philosophy becomes ideology if something is taken for granted because a particular person said it. Nothing is correct by virtue of meeting Gödel's assessment.
    – Philip Klöcking ♦
    Oct 18, 2019 at 21:22
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    I suppose mentioning Torkel’s name and book was somehow a stronger claim?
    – Incompleteness
    Oct 24, 2019 at 23:38

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