As the title explains, I'm trying to solve the following exercise that was left for the reader in my lecture notes:
Show that for any two formulae $F(v_1)$ and $G(v_1)$ in $L_E$ (language of arithmetic with exponentiation) with one free variable, there exist sentences $X$ and $Y$ such that the sentences $(X ↔ G(\overline{⌜Y⌝}))$ and $(Y ↔ F(\overline{⌜X⌝}))$ are both true.
(where $⌜X⌝$ means the Gödel number of X, and $\overline{⌜X⌝}$ is the 'logical equivelent' of that Gödel number.)
This clearly resembles the diagonal lemma/fixed-point theorem, except instead of one self-referential sentence we're supposed to have two sentences referring to each other.
I've tried playing around with different combinations of what you might take $X$ and $Y$ to be, attempting to adapt the proof of the diagonal lemma, but haven't had much luck.
Any suggestions anyone could offer would be much appreciated.
[if I've missed out any important details, let me know and I'll add them1]