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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?

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    $\begingroup$ I'm guessing you mean to ask for an answer along the lines similar to the result about the completeness theorem, e.g. mathoverflow.net/questions/68335/… $\endgroup$
    – David Roberts
    Jun 5, 2013 at 5:06
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    $\begingroup$ 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 mathoverflow.net/howtoask, 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, $\endgroup$ Jun 5, 2013 at 6:56
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    $\begingroup$ Or at least, spell Gödel's name correctly... $\endgroup$
    – Zhen Lin
    Jun 5, 2013 at 7:27
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    $\begingroup$ These are fair points. @Bauer: I did mean qualifier not quantifier, as an adjective qualifies. $\endgroup$ Jun 5, 2013 at 11:03
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    $\begingroup$ arxiv.org/abs/math/0305282 $\endgroup$
    – uhbif19
    Mar 26, 2015 at 22:51

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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.

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    $\begingroup$ Your assumption is correct, this is close to what I was looking for! $\endgroup$ Jun 5, 2013 at 12:27
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    $\begingroup$ Joyal's proof (or at least a proof very much in the same spirit) has been written up last year: arxiv.org/pdf/2004.10482.pdf $\endgroup$ Apr 13, 2021 at 14:57
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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.

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  • $\begingroup$ I think thats only a categorical formulation of the diagonal lemma there. $\endgroup$ Jun 15, 2013 at 0:56
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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.

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Lawvere's fixpoint theorem generalizes the diagonal argument, and the incompleteness theorem can be taken as a special case.

The proof can be found in Frumin and Massas's Diagonal Arguments and Lawvere's Theorem. Here is a copy.

Definition: A morphism $f: X\times X\to Y$ is weakly point-surjective iff for every $g: X\to Y$, there is a $t: 1\to X$ such that, for all $x: 1\to X$: $$gx=f\langle x,t\rangle$$

Lawvere's Fixpoint Theorem:

Let $\mathbf{C}$ be a category with a terminal object and binary products. If $f: X\times X\to Y$ is weakly point-surjective, then every morphism $\alpha: Y\to Y$ has a fixpoint $y: 1\to Y$.

Consider a first-order theory $\mathrm{T}$. We form $\mathbf{C}_\mathrm{T}$ a classifying category of $\mathrm{T}$ in the following way:

The $\mathbf{C}_\mathrm{T}$-objects are generated by a sort object $A$ (more object if the theory is multi-sorted), and an object $2$.

The $\mathbf{C}_\mathrm{T}$-morphisms are equivalence classes of (tuples of) formulas $A^n\to 2$ or terms $A^n\to A$ of $\mathrm{T}$.

In particular, morphisms $1\to 2$ are sentences, and morphisms $1\to A$ are constant terms.

A theory is complete iff $\operatorname{Hom}(1,2)=\{\top,\bot\}$.

  • Undefinability of $\operatorname{sat}$. Suppose that the satisfiabilty predicate is definable in $\mathrm{T}$: $\vdash \operatorname{sat}(a,\ulcorner\varphi\urcorner)\leftrightarrow\varphi(a)$ for all $\varphi,a$.

In categorical terms, we have a Godel encoding $\ulcorner \urcorner: \operatorname{Hom}(A^n,2)\to \operatorname{Hom}(1,A)$, and a formula $\operatorname{sat}: A^2\to 2$, such that for $\varphi: A\to 2$ and $a: 1\to A$, $\operatorname{sat}\langle a,\ulcorner\varphi\urcorner\rangle=\varphi a$.

But this is exactly the condition for weak point-surjectivity!

  • Undefinability of $\operatorname{truth}$. Suppose that $\mathrm{T}$ has a 'truth' predicate: $\operatorname{true}\circ\ulcorner\varphi\urcorner=\varphi$ for all $\varphi\in \operatorname{Hom}(1,2)$.

Suppose that $\mathrm{T}$ supports 'substitution': $\operatorname{subst}\langle a,\ulcorner\varphi\urcorner\rangle=\ulcorner\varphi(a)\urcorner$.

Then we can define $\operatorname{sat} := \operatorname{true}\circ \operatorname{subst}$.

  • Incomplenteness. Suppose that 'provability' is representable in $\mathrm{T}$: $\mathrm{T}\vdash\varphi\iff\mathrm{T}\vdash\operatorname{prov}(\ulcorner\varphi\urcorner)$.

If $\mathrm{T}$ is complete, then $\varphi=\top$ or $\varphi=\bot$.

And $\varphi=\top\implies \operatorname{prov}(\ulcorner\varphi\urcorner)=\top$, $\varphi=\bot\implies \operatorname{prov}(\ulcorner\varphi\urcorner)=\bot$.

Therefore, for all $\varphi\in \operatorname{Hom}(1,2): \operatorname{prov}\circ\ulcorner\varphi\urcorner=\varphi$, i.e. $\operatorname{true}$ is $\operatorname{prov}$.

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