# a stronger fixed point theorem

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.

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.