If I have a sentence $S$ over $\mathrm{PA}$ (or some similar theory which can encode FOL), I can look at the set $X_S$ which contains $S$ and is closed under the standard logical connectives as well as the provability predicate. So $X_S$ contains sentences like '$S$ is true and PA proves that PA does not prove that $S$ is false'.

How many of these sentences can be shown to be logically equivalent, say, in the context of ZFC? In other words, how many sentences about all meta truth values concerning $S$ can be formed that don't reduce to some simpler statement?

I already read about $\Sigma_1$-soundness, which is is not provable in PA, but I didin't get whether it's provable in ZFC, just like consistency of PA isn't provable in PA itself but in ZFC, as far as I can remember.


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