What is the exact, formal statement of Gödel's first incompleteness theorem?

Question

I am looking for the explicit formal statement of Gödel's first incompleteness theorem in a formal language (which I assume is the language of first-order Peano arithmetic), permitting only the addition of abbreviating symbols $G(\cdot)$ (used in place of $\ulcorner\cdot\urcorner$) and $G^{-1}(\cdot)$ (the reason for using $G(\cdot)$ instead of $\ulcorner\cdot\urcorner$), and $\mathsf{prf}_T(\cdot)$ - $T$ being the theory whose language the theorem is stated in, provided it is not Peano arithmetic.

Such a statement should begin either $\exists x$, or $T\vdash$.

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Response to Noah Schweber's Answer:

Provided that $T\vdash\mathsf{prf}_T(\ulcorner\phi\urcorner)\to\phi$, if $T\vdash \neg\mathsf{prf}_T(\ulcorner\bot\urcorner)\to\exists x(\neg\mathsf{prf}_T(x)\land\neg\mathsf{prf}_T(\mathsf{neg}(x)))$, then $T\vdash \exists x(\neg\mathsf{prf}_T(x)\land\neg\mathsf{prf}_T(\mathsf{neg}(x)))$. In other words, incompleteness conditional the assumption of consistency entail incompleteness. The proof for the case of $\mathsf{PA}$ is as follows:

$$\begin{array} \ 1. & \quad\neg\mathsf{prf_{PA}}(\ulcorner 0=S0\urcorner)\to\exists x(\neg\mathsf{prf_{PA}}(x)\land\neg\mathsf{prf_{PA}}(\mathsf{neg}(x))) & \mathsf{IncThm}\ \small(given)\\ 2. & \quad \neg(0=S0) &\mathsf{Thm1}\\ 3. & \quad\qquad\mathsf{prf_{PA}}(\ulcorner0=S0\urcorner) &\text{assumption}\\ 4. & \quad\qquad\mathsf{prf_{PA}}(\ulcorner0=S0\urcorner)\to(0=S0) & \mathsf{T_{PA}}\\ 5. & \quad\qquad 0=S0 &\mathsf{MP}\ 4,3\\ 6. & \quad\qquad \bot & \mathsf{\bot I}\ 5,2\\ 7. & \quad\qquad \exists x(\neg\mathsf{prf_{PA}}(x)\land\neg\mathsf{prf_{PA}}(\mathsf{neg}(x))) & \mathsf{\bot E}\ 6\\ 8. & \quad \mathsf{prf_{PA}}(\ulcorner0=S0\urcorner)\to\exists x(\neg\mathsf{prf_{PA}}(x)\land\neg\mathsf{prf_{PA}}(\mathsf{neg}(x))) & \mathsf{\to I}\ 3-7\\ 9. & \quad \exists x(\neg\mathsf{prf_{PA}}(x)\land\neg\mathsf{prf_{PA}}(\mathsf{neg}(x))) & \mathsf{Thm}2 \end{array}$$

The last line of the proof is a case of the derived inference rule $\phi\to\psi,\neg\phi\to\psi\vdash\psi$, referred to here as $\mathsf{Thm}2$. The proofs of $\mathsf{Thm}1$ and $\mathsf{Thm}2$ are left as exercises for the reader™.

I've also decided $\mathsf{prf}_T(\ulcorner\phi\urcorner)\to\phi$ is a realization of the Modal axiom $\mathsf{T}$, usually expressed as $\Box\phi\to\phi$, within the theory $T$, hence the justification in line $3$ being labelled $\mathsf{T_{PA}}$. There's probably something to be said about provability logic here, but I'm not remotely qualified to say it.

Now perhaps it is the case that $T\nvdash\mathsf{prf}_T(\ulcorner\phi\urcorner)\to\phi$, in which case... well, I'm not sure. Otherwise, maybe a stronger condition is needed to avoid having a theory proving its own incompleteness? My preliminary guess would be to have something like...

$$\forall x(\mathsf{prf}_T(x)\leftrightarrow\neg\mathsf{prf}_T(\mathsf{neg}(x)))$$

...replace the antecedent.

Answer 1

There are actually a few different results (basically different levels of generality of the same idea) collectively called the first incompleteness theorem, e.g. "$\mathsf{PA}$ is undecidable if $\omega$-consistent," "$\mathsf{PA}$ is undecidable if consistent," "Every computably axiomatizable theory interpreting $\mathsf{PA}$ is incomplete if consistent," etc.

That said, fixing some "reasonable" theory $T$, here is a true statement which is one of the forms of the first incompleteness theorem:

If $T$ is consistent then $T$ is incomplete.

And here's a version of the above which is explicitly arithmetic, modulo a couple abbreviations, and provable in (say) $\mathsf{PA}$:

$[\neg \mathsf{prf}_T(\mathfrak{x})]\rightarrow\exists x(\neg \mathsf{prf}_T(x)\wedge\neg \mathsf{prf}_T(\mathsf{neg}(x))$.

The abbreviations are as follows:

• "$\mathsf{prf}_T$" is the usual provability predicate for $T$.

• "$\mathfrak{x}$" is the numeral corresponding to the Godel number of the logical constant for contradiction, $\perp$ (or some fixed contradiction like "$0=1$" if you don't include $\perp$ in your logical apparatus, but it's convenient to do so so I will).

• "$\mathsf{neg}$" is the canonical definition of the primitive recursive function sending the Godel number of a sentence to the Godel number of that sentence's negation. This is an important component of the argument, and is sufficiently tedious to write down in detail that including a dedicated abbreviation for it too is a good idea. I would caution against thinking of this as (using your notation) "$G(\neg G^{-1}(x))$" - there are serious issues with using "un-Godel numbering" $G^{-1}(\cdot)$ if you're not very careful. Instead, $\mathsf{neg}$ is just some complicated arithmetical operation which happens to have the property that if $\varphi$ is a sentence with Godel number $k$ then $\neg\varphi$ has Godel number $\mathsf{neg}(k)$.

(Additionally I'm assuming that we're using a surjective Godel numbering system, so we don't have to include a clause after the "$\exists x$" saying that $x$ is in fact a Godel number.)

In particular, this works for $T=\mathsf{PA}$, in the following sense:

$\mathsf{PA}\vdash[(\neg \mathsf{prf}_\mathsf{PA}(\mathfrak{x}))\rightarrow\exists x(\neg \mathsf{prf}_\mathsf{PA}(x)\wedge\neg \mathsf{prf}_\mathsf{PA}(\mathsf{neg}(x))]$.

Note that this does not mean that $\mathsf{PA}$ proves its own incompleteness (which would lead to an inconsistency in $\mathsf{PA}$ per the second incompleteness theorem); rather, $\mathsf{PA}$ proves its own incompleteness conditional on the assumption of its consistency, which is a much weaker thing.

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The "big" version of the first incompleteness theorem of course is even stronger:

$\mathsf{PA}$ proves that every consistent primitive recursively axiomatizable theory interpreting $\mathsf{Q}$ is incomplete.

(And even this can be strengthened, but ignore that for now.)

This is the following:

$\mathsf{PA}\vdash\forall e[\mathsf{int_Q}(e)\wedge\neg\mathsf{prprf}(e,\mathfrak{x})\rightarrow \exists x(\neg\mathsf{prprf}(e,x)\wedge\neg\mathsf{prprf}(e,\mathsf{neg}(x)))].$

Here $\mathfrak{x}$ and $\mathsf{neg}$ are as above. The new ingredients are the unary relation "$\mathsf{int_Q}$" and the binary provability relation $\mathsf{prprf}$. The former is read as "$e$ is a primitive recursive index for a theory interpreting $\mathsf{Q}$," which is extremely tedious to write out. The latter is the "uniform" provability predicate; intuitively, read "$\mathsf{prprf}(e,x)$" as "$x$ is the Godel number of a theorem in the primitive recursively axiomatized theory with index $e$." When we build the provability predicate for (say) $\mathsf{PA}$, we really go through all the work of building $\mathsf{prprf}$ itself, so this isn't actually adding any complexity.