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Given a unary predicate $\phi$, the Fixed Point Lemma of PA tells us that there is a sentence $S$ such that:

$$\mbox{PA} \vdash S \leftrightarrow \phi (\ulcorner S \urcorner)$$

(Note that $\phi$ doesn't even have to be in the language of arithmetic - $\phi$ could, for example, be some sort of truth predicate T, as to prove the Fixed Point Lemma there is no requirement that induction hold of predicates involving $\phi$.)

My question is whether one can say something stronger: namely, there is a sentence $S$ such that $S$ and $\phi(\ulcorner S \urcorner)$ are literally (i.e., syntactically) the same sentence.

I have heard people talk as if this also holds, but I don't see how the usual proof of the Fixed Point Lemma gives this.

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This was discussed at MO (and so I've made this answer CW). Briefly, some but not all Godel numberings admit such "strong fixed points."

It's easy to verify that none of the usual Godel numberings admit strong fixed points: just check that in any of the usual numbering the Godel number of the numeral of $k$ is greater than $k$ for every natural number $k$. So numbering systems which do admit strong fixed points have to be a bit complicated. The existence of such a numbering was originally claimed by Kripke without proof; a proof appears in Visser's $2002$ paper Semantics and the Liar Paradox.

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