MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A significant result in set theory was shown by Cohen when he showed that the continuum hypothesis was independent of ZFC using a new technique called forcing. In Topos theory, this result has a new geometric interpretation: the sheaf for the dense topology on the poset of finite approximations on the 'impossible monic' form a new model of sets where this monic is actually there.

Now, presumably Gödel's incompleteness theorem remains valid for typed intuitionistic higher-order logic; such a logic is the internal logic of a topos.

Is there a categorical proof of Gödel's theorem in Topos theory? Does Gödel's theorem say anything geometric or throw new light on the theorem when interpreted in a topos?

share|cite|improve this question
I'm guessing you mean to ask for an answer along the lines similar to the result about the completeness theorem, e.g.… – David Roberts Jun 5 '13 at 5:06
Dear Mozibur, This question could be improved with some rewriting. You should provide some background, to clarify your question. You should also sharpen your question: "what does A mean for B" is very broad, and does not make a good MathOverflow question. As I'm sure you saw on, an extremely important thing for all questions is to "do your homework", but right now your question looks like idle speculation with no homework done. I am far from an expert, but I expect there is some meat here. You won't find it as the question stands, I'm afraid. All the best, – Theo Johnson-Freyd Jun 5 '13 at 6:56
Or at least, spell Gödel's name correctly... – Zhen Lin Jun 5 '13 at 7:27
These are fair points. @Bauer: I did mean qualifier not quantifier, as an adjective qualifies. – Mozibur Ullah Jun 5 '13 at 11:03
@Roberts: Thanks. Thats the kind of result I was looking for. – Mozibur Ullah Jun 5 '13 at 11:04
up vote 15 down vote accepted

This is not exactly what you asked for but I think it's reasonably close to what you want...

The idea of recasting Gödel's results in the context of category theory has led André Joyal to develop arithmetic universes, a minimalistic category tailored for that purpose. Unfortunately, Joyal never published this as explained by Paul Taylor in this recent answer.

share|cite|improve this answer
Your assumption is correct, this is close to what I was looking for! – Mozibur Ullah Jun 5 '13 at 12:27

It means that higher-order intuitionistic logic with natural numbers and Heyting arithmetic cannot prove its own consistency.

I think it is hard to see from your question what sort of an answer you are expecting. Are you looking for topos-theoretic formulations of Gödel's theorems? Or the impact that the theorems have on topos theory? I can amend the answer once I understand what you are looking for.

share|cite|improve this answer

There is a reformulation in categorical terms of Godel's incompleteness theorem in the book "Conceptual mathematics" by Schanuel and Lawvere. There are also notes of Gromov Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1, 2 where on page 16-17 he discusses "an adaptation of [Schanuel-Lawvere] argument". For both approaches the key is Cantor's diagonal argument; I do not think either mentions topos theory.

share|cite|improve this answer
I think thats only a categorical formulation of the diagonal lemma there. – Mozibur Ullah Jun 15 '13 at 0:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.