Minimal arithmetic proving a statement similar to a Gödel sentence

Question

I will use $\textbf{Q}$ to denote minimal arithmetic for this post. (I suppose Robinson arithmetic would also suffice (?))

Suppose we have $F(x)$ be a formula defining, in $\textbf{Q}$ the primitive recursive relation "$x$ is a sentence of the language of arithmetic". Let $G$ be such that

$\textbf{Q}\vdash G \leftrightarrow F(\ulcorner G \urcorner)$

I am trying to determine whether or not $\textbf{Q} \vdash G$.

Answer 1

Here's how I see this problem (developed somewhat from my comment above): suppose we have $Q \vdash G \leftrightarrow F(\ulcorner G \urcorner)$. Clearly $G$ is a sentence, so $F(\ulcorner G \urcorner)$ holds (i.e. is true in the standard model).

As you point out, the relation "$x$ is a sentence of the language of arithmetic" is primitive recursive, so it has a $\Sigma_0$-formula expressing it. If $F$ is such a $\Sigma_0$ formula, then since $Q$ is $\Sigma_0$-complete, $Q \vdash F(\ulcorner G \urcorner)$, so by the biconditional above $Q \vdash G$.

However, if $F$ was not chosen to be $\Sigma_0$, I don't think we can automatically assume that $Q \vdash F(\ulcorner G \urcorner)$.