Tarski's undefinability theorem

Remarks

Proof idea: suppose $\mathrm{True}(x)$ exists. The diagonal lemma, applied in the theory of all sentences true in $\mathcal{N}$, gives a sentence $\lambda$ with $\mathcal{N} \models \lambda \leftrightarrow \neg \mathrm{True}(\ulcorner \lambda \urcorner)$. Then $\lambda$ is true exactly when it is not true, a contradiction. This is a precise form of the liar paradox.

By contrast, for each $n$ the set of true $\Sigma_n$ sentences is definable in $\mathcal{N}$; only full truth is not. A syntactic form also holds: if $T$ is a consistent theory that contains Robinson arithmetic, there is no formula $\mathrm{True}(x)$ with $T \vdash \mathrm{True}(\ulcorner \varphi \urcorner) \leftrightarrow \varphi$ for every sentence $\varphi$.

Going deeper

For a student

Compare with Gödel's incompleteness theorems: provability in a computably axiomatized theory is definable in arithmetic, but truth is not. So truth and provability differ, which gives another proof of the first incompleteness theorem. In terms of the arithmetical hierarchy, the set of true sentences is not $\Sigma^0_n$ for any $n$.