Let's consider an arithmetic theory such as Peano Arithmetic. Suppose the Gödel codes of all the true sentences are even and all the false sentences are odd. Then, how can there be a fixed point for the predicate, $\phi(x)=$"x is odd."?
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6"Suppose the Gödel codes of all the true sentences are even and all the false sentences are odd." This can't occur, pretty much for exactly the reason you describe - this is Tarski's undefinability theorem. Not every seemingly-plausible property of Godel codings is actually realizable! (This raises the question of what exactly we mean by "Godel coding," and in fact we can give a precise if somewhat weird definition; basically, a Godel coding is a function from sentences to numbers satisfying a couple definability properties ... – Noah Schweber Jun 16 '22 at 01:28
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The fixed point theorem is a consequence of these definability properties, and meanwhile no such function can map all true statements to even numbers and all false statements to odd numbers.) – Noah Schweber Jun 16 '22 at 01:31
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Or, conversely, if we do take some coding (which necessarily won't be a Gödel coding in @NoahSchweber's sense), then we'll have trouble doing things that we would want to be able to do with the coding, such as writing $\phi(x,n,t)={}$"the $n$th character of the statement coded by $x$ is the character coded by $t$" (in other words, word-processing operations) – Akiva Weinberger Jun 16 '22 at 02:02
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We might as well ask, "suppose all the programs of halting (zero-input) algorithms end in 0, and all the programs of non-halting algorithms end in 1." (I'm referencing the halting problem here.) Reading and executing those programs becomes an uncomputable task. – Akiva Weinberger Jun 16 '22 at 02:14
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Turning my comments into an answer, the issue is that when you write
Suppose the Gödel codes of all the true sentences are even and all the false sentences are odd
you're making a very false assumption: that that situation can in fact occur with respect to a Godel coding system. In fact, the rest of your question is more-or-less the proof that this can't happen! Keep in mind that the diagonal lemma is not trivial; it doesn't apply to all functions from formulas to numbers, only to some. So the fact that this places restrictions on the possible behaviors shouldn't be a surprise.
Specifically, we have the following result:
(Tarski's undefinability theorem) Suppose $\mathscr{F}$ is an injective function from formulas to numbers such that the "substitution relation" $$\mathfrak{S}:=\{(l,m,n)\in\mathbb{N}^3: l\in ran(\mathscr{F})\wedge \mathscr{F}(\mathscr{F}^{-1}(l)[\underline{m}/x])=n\}$$ is definable in $\mathcal{N}$. Then $\mathscr{F}[Th(\mathcal{N})]$ is not definable in $\mathcal{N}$.
That's a lot of notation! Here's what it means:
"Formula" means "first-order formula in the language of arithmetic," and "number" means "natural number."
$\underline{n}$ is the numeral corresponding to $n$, e.g. $\underline{3}=(1+1)+1$.
$\varphi[t/x]$ is the formula gotten from $\varphi$ by replacing all free occurrences of the variable $x$ with the term $t$.
So intuitively, $\mathfrak{S}(l,m,n)$ means "If we take the formula with code $l$ and substitute $m$ for the variable $x$, the resulting formula has code $n$." And it's not hard to see that we really lean on the definability of $\mathfrak{S}$ in proving the diagonal lemma for $\mathscr{F}$, when in fact $\mathscr{F}$ is a "reasonable coding system," so this matters.
Continuing on:
"Definable" means what it usually does in model theory.
$\mathcal{N}$ is the structure $(\mathbb{N};+,\times,0,1)$ (or anything "morally equivalent" to that).
$\mathscr{F}[Th(\mathcal{N})]$ is the image under $\mathscr{F}$ of the theory of $\mathcal{N}$; or, more intuitively, the set of $\mathscr{F}$-codes of true sentences.
Noah Schweber
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A lot of Gödel's stuff is basically just about word processors, because checking whether a proof is valid is essentially a word processing task. First (arithmetization) you show that the arithmetic theory can represent word processing stuff, and second (diagonalization) you do Quine-esque stuff to write a sentence that refers to its own provability. (In one proof, anyway.) – Akiva Weinberger Jun 16 '22 at 02:11