Fix a Gödel numbering scheme, and let $\operatorname{Thm}_{\mathsf{PA}}$ be the corresponding numerical provability predicate for Peano arithmetic. Suppose $\theta$ and $\xi$ are two sentences in arithmetic such that $\mathsf{PA} \vdash \theta \leftrightarrow \neg \operatorname{Thm}_{\mathsf{PA}} ( \ulcorner \theta \urcorner)$ and $\mathsf{PA} \vdash \xi\leftrightarrow \neg \operatorname{Thm}_{\mathsf{PA}} ( \ulcorner \xi\urcorner)$. Why must it be the case that $\mathsf{PA} \vdash \theta \leftrightarrow \xi$?
A few notes:
- The diagonal lemma asserts existence but not uniqueness, so the usual construction of the Gödel sentence is non-unique.
- The key difference with this post is that we're considering a fixed Gödel numbering scheme throughout.
- This is an exercise in Leary and Kristiansen's Friendly Introduction to Mathematical Logic (chapter 6.6 exercise 2).
Could the answer be as simple as this?
I'm paraphrasing Note 2 from here. Every Godel Sentences is equivalent to the statement $Con_{PA}$, where $Con_{PA}$ is defined as:
$$Con_{PA} = \neg\exists x \;\; Thm_{PA}(\ulcorner x \urcorner) \;\land\; \neg Thm_{PA}(\ulcorner x \urcorner)$$
$Con_{PA}$ doesn't depend on the particular Godel Sentence, so it must be equivalent to both $\theta$ and $\xi$. Therefore: $$\theta \leftrightarrow Con_{PA} \leftrightarrow \xi$$
And so:
$$PA \vdash \theta \leftrightarrow \xi$$