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$\Sigma_1$-sound and computable first-order theory, which is stronger than $\mathsf{PA}^-$, is incomplete.
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Let $T$ be a $\Delta_1$-definable arithmetic theory, stronger than $\mathsf{R}_0$ and $\Sigma_1$-sound.
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```lean
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theorem LO.FirstOrder.Arith.first_incompleteness
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(T : LO.FirstOrder.Theory ℒₒᵣ)
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[DecidablePred T]
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[𝐄𝐐 ≼ T]
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[𝐏𝐀⁻ ≼ T]
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[LO.FirstOrder.Arith.SigmaOneSound T]
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[LO.FirstOrder.Theory.Computable T] :
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¬LO.System.Complete T
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```
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This theorem is proved two distinct approach.
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-[G1 in `FirstIncompleteness.lean`](https://formalizedformallogic.github.io/Foundation/docs/Logic/FirstOrder/Incompleteness/FirstIncompleteness.html#LO.FirstOrder.Arith.first_incompleteness)
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-[G1 in `SelfReference.lean`](https://formalizedformallogic.github.io/Foundation/docs/Logic/FirstOrder/Incompleteness/SelfReference.html#LO.FirstOrder.Arith.FirstIncompletenessBySelfReference.not_complete)
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`FirstIncompleteness.lean` is computability theoretic proof, while `SelfReference.lean` uses a well-known self-referential argument.
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### G1 in `FirstIncompleteness.lean`
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Define a set of formulae with one variable.
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$$ D \coloneqq \{\varphi \mid T \vdash \lnot \varphi({\ulcorner \varphi \urcorner}) \} $$
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$D$ is r.e. since $T$ is computable. (one could use Craig's trick to weaken this condition to r.e., but I will not do that here.)
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By the representation theorem, there exists a $\Sigma_1$ formula $\rho(x)$ that represents $D$. i.e.,
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$$ T \vdash \rho({\ulcorner \varphi \urcorner}) \iff T \vdash \lnot \varphi({\ulcorner \varphi \urcorner})$$
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Let $\gamma := \rho({\ulcorner \rho \urcorner})$. The next follows immediately.
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$$ T \vdash \gamma \iff T \vdash \lnot \gamma $$
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Thus, as $T$ is consistent, $\gamma$ is undecidable from $T$.
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### Representeation Theorem
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###G1 in `SelfReference.lean`
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#### Theorem: Let $S$ be a r.e. set. Then, there exists a formula $\varphi_S(x)$ such that $n \in S \iff T \vdash \varphi_S(\overline{n})$ for all $n \in \mathbb{N}$.
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Since the substitution of a formula is computable, there exists an $\Sigma_1$ formula $\mathrm{ssbs}(x, y, z)$ that represents this:
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$$
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T \vdash (\forall x)[\mathrm{ssbs}(x, {\ulcorner \varphi \urcorner}, {\ulcorner \psi \urcorner})
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\leftrightarrow x = {\ulcorner \varphi({\ulcorner \psi \urcorner}) \urcorner}]
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$$
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Define a sentence $\mathrm{fixpoint}_\theta$ for formula (with one variable) $\theta$ as follows.
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```lean
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lemma re_complete
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[𝐑₀ ≼ T] [Sigma1Sound T]
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{p : ℕ → Prop} (hp : RePred p) {x : ℕ} :
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p x ↔ T ⊢! ↑((codeOfRePred p)/[‘↑x’] : Sentence ℒₒᵣ)
Let $G := \mathrm{fixpoint}_{\lnot\mathrm{prov_T}(x)}$ (Gödel sentence; the sentence that states "This sentence is not provable"),
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where $\mathrm{prov}_T(x)$ is a formula represents provability.
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Since $G$ is undecidable, this results in the incompleteness of $T$.
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- Assume that $T \vdash G$. $T \vdash \lnot \mathrm{prov}_T(\ulcorner G \urcorner)$ follows from the fixpoint lemma,
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while $T \vdash \mathrm{prov}_T(\ulcorner G \urcorner)$ follows from the hypothesis. a contradiction.
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- Assume that $T \vdash \lnot G$. $T \vdash \mathrm{prov}_T(\ulcorner G \urcorner)$ follows from the fixpoint lemma,
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and $T \vdash G$ follows. a contradiction.
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Define a set of formulae with one variable.
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$$ D \coloneqq \{\varphi \mid T \vdash \lnot \varphi({\ulcorner \varphi \urcorner}) \} $$
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$D$ is r.e. since $T$ is $\Delta_1$-definable. (one could use Craig's trick to weaken this condition to $\Sigma_1$, but I will not do that here.)
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By the representation theorem, there exists a $\Sigma_1$ formula $\rho(x)$ that represents $D$. i.e.,
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$$ T \vdash \rho({\ulcorner \varphi \urcorner}) \iff T \vdash \lnot \varphi({\ulcorner \varphi \urcorner})$$
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Let $\gamma := \rho({\ulcorner \rho \urcorner})$. The next follows immediately.
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$$ T \vdash \gamma \iff T \vdash \lnot \gamma $$
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Thus, as $T$ is consistent, $\gamma$ is undecidable from $T$. ∎
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## About Second Theorem
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To prove the second incompleteness theorem, as outlined in Gödel's original paper, one can derive it by proving the first incompleteness theorem again within arithmetic.
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Although this fact has not yet been formalized in this project.
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These efforts are being undertaken in a separated repository [iehality/Arithmetization](https://github.com/iehality/Arithmetization).
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Notably, the formalization of the second incompleteness theorem has already been accomplished by L. C. Paulson in Isabelle.
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However, owing to the technical simplicity for coding and others. this formalization is on _hereditarily finite sets_ and not on arithmetic.
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-[L. C. Paulson, "A machine-assisted proof of Gödel's incompleteness theorems for the theory of hereditarily finite sets"](https://www.repository.cam.ac.uk/items/bda52431-26e0-4e86-8d63-409bcedd4617)
#### Gödel's Second Incompleteness Theorem: $T$ cannot prove its own consistency, i.e., $T \nvdash \mathrm{Con}_T$ if $T$ is consistent. Moreover, $\mathrm{Con}_T$ is undecidable from $T$ if $\mathbb{N} \models T$.
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#### Theorem: $T$ cannot prove its own consistency, i.e., $T \nvdash \mathrm{Con}_T$ if $T$ is consistent. Moreover, $\mathrm{Con}_T$ is undecidable from $T$ if $\mathbb{N} \models T$.
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