"This statement is false" but with Gödel numbering

Question

I know that one cannot construct the paradoxical sentence

This statement is false

with the formal language of logic, since there is no way to denote the proposition itself within the statement, or otherwise it would not be a well-formed formula (with something like $p := \lnot p$). And this is why it is generally not considered as a proposition. (See "This statement is false.")

However, after reading about Gödel's first incompleteness theorem, in which Gödel used the technique of Gödel numbering, such that one can construct well-formed statements which "takes references" of other statements by indexing. For example, Gödel's proof involved the proposition:

The statement with Gödel's number $g$ has no proof

Therefore, is it possible to turn "this statement is false", by the same way with Gödel numbering, into some well-formed propositions like

The statement with Gödel's number $n$ is false

which is Gödel coded with exactly the number $n$?

Answer 1

In Gödel, you can only say if something has a proof or not, not whether a statement is “true” or “false” in some other sense.

So you can say:

This statement can be proven false.

Or:

This statement cannot be proven true.

Neither one is a paradox, because of the possibility of undecidable statements.

Answer 2

If we really want to talk about truth (not provability, as has been suggested as a "backup"), the liar paradox gives us a no-go theorem. If we could express "$n$ is the Gödel number of a true sentence in the language of arithmetic" as a predicate $T(n)$ in the language of arithmetic, then we could express the liar paradox by using diagonal lemma to find a sentence $S$ such that $S$ holds if and only if $\lnot T(\ulcorner S\urcorner)$ holds. So such a predicate $T$ cannot exist. This is Tarski's theorem, Gödel's theorem's underrated cousin.

Edit Also, an obligatory mention that I failed to fulfill in the initial answer:

We can, in a precise sense, "almost" express arithmetical truth in arithmetic. For any natural number $k$, there is an arithmetic predicate $T_k(n)$ which holds if and only if $n$ is the Gödel number of a true sentence with no more than $k$ quantifiers. But, crucially, this is a different formula for each $k$, not a single formula in two variables $T(k,n),$ so the obvious idea to express truth for arbitrary sentences as $"\forall k T_k(n)"$ does not work.

Answer 3

No, this can't happen - this is due to Tarski.

The language around Godel's theorems often makes the whole apparatus of "Godelization" seem magical in how $(i)$ properties of formulas are translated into properties of numbers and $(ii)$ self-reference is somehow made possible. I understand the desire to sweep details under the rug here, but I think this is actually one situation where this makes things harder to understand; in particular, without those details at least being hinted at, it is indeed reasonable to wonder whether we can somehow "Godelize the liar paradox."

The short version is that we don't get "expressibility for free:" any particular way of representing formulas by numbers is going to have some strengths and some weaknesses, and in fact there is no such representation scheme which simultaneously lets us perform self-reference and talk about truth (so the liar paradox cannot emerge).

I've taken a rather abstract approach below, but please don't be alarmed - I think that starting this way will help avoid false starts in terms of intuition. The one logical notion that I'll lean on here is that of definability in a structure; if you're unfamiliar with this, see e.g. this old answer of mine.

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Let's say that a numbering is any injective function $\mathfrak{F}:Form\rightarrow\mathbb{N}$, where $Form$ is the set of formulas in the language of arithmetic. I really do mean any (injective) function here; numberings, in the context of this answer at least, are not required to be "nice" in any way whatsoever.

Now a priori the structure $\mathcal{N}=(\mathbb{N};+,\times)$ can only directly talk about natural numbers. However, every numbering $\mathfrak{F}$ lets us "port over" properties of formulas to properties of natural numbers: given a numbering $\mathfrak{F}$ and a set of formulas $S\subseteq Form$, say that $S$ is definable via $\mathfrak{F}$ iff $$S^\mathfrak{F}:=\{\mathfrak{F}(\sigma):\sigma\in S\}$$ is a definable set in $\mathcal{N}$ in the usual sense. We lift this to higher-arity relations on $Form$, functions from $Form$ to $Form$, etc. in the obvious ways. This is a very loose notion - every set of formulas (or function of formulas, or etc.) is definable via some numbering!

However, most numberings are really really silly. Things get more interesting when we restrict attention to numberings which have already some "basic defining power:" for example, one very low-level requirement should be that the function $$\mathsf{CONJ}: Form\times Form\rightarrow Form: (\varphi,\psi)\mapsto\varphi\wedge\psi$$ should be definable via $\mathfrak{F}$ in the sense that the relation $$\mathsf{CONJ}^\mathfrak{F}:=\{(x,y,z)\in\mathbb{N}^3: x,y,z\in ran(\mathfrak{F})\mbox{ and }\mathsf{CONJ}(\mathfrak{F}^{-1}(x),\mathfrak{F}^{-1}(y))=\mathfrak{F}^{-1}(z)\}$$ should be definable in $\mathcal{N}$. Taking two formulas and forming their conjunction is intuitively pretty easy, so we really ought to be able to "Godelize" that process if our numbering isn't totally stupid.

Being vague in the interest of brevity, say that a numbering is nice if a few appropriate operations and sets are definable via it. The important thing is that nice numberings are really easy to construct, to the point that you're highly unlikely to whip up a non-nice numbering by accident. It turns out that niceness, while very mild on the face of it, has surprisingly strong consequences. In my opinion, simultaneously the most important and most surprising is the following:

Suppose $\mathfrak{F}$ is a nice numbering. Then for every first-order formula $\varphi(x)$ in the language of arithmetic there is a sentence $\theta$ such that $$\mathcal{N}\models \theta\leftrightarrow \varphi(\underline{\mathfrak{F}(\theta)}),$$ where $\underline{n}$ is the numeral corresponding to $n$.

Basically, nice numberings always make self-reference possible! This is one "flavor" of the diagonal lemma.

We can quickly apply this to get a restriction on the "definability power" of a nice numbering (which is interesting since niceness is a purely positive property on the face of things): specifically, if $\mathfrak{F}$ is a nice numbering then the set of false sentences is not definable via $\mathfrak{F}$. Otherwise we'd be able to produce a sentence $\theta$ such that $\mathcal{N}\models\theta\leftrightarrow\mathcal{N}\models\neg\theta$, which is a contradiction.

As an aside, Godel's incompleteness theorem involves a more detailed analysis: we need to bring provability (relative to a fixed "reasonable" theory) into the picture, and look at not just what is true in the structure $\mathcal{N}$ but also what is or is not provable. The general flavor remains the same, but there is additional technical overhead.