Given a φ independent of PA which is true in the standard model, will always (PA+ ¬φ) be Σn-unsound for some n?

Question

Given a φ independent of PA which is true in the standard model, will always (PA+ ¬φ) be Σn-unsound for some n? This is a follow up from a previous question: Given a φ independent of PA which is true in the standard model, will always (PA+ ¬φ) be ω-inconsistent?.

Answer 1

Yes. Every formula "is" $\Sigma^0_n$ for sufficiently large $n$, and the rest follows directly from the definition of a theory being $\Sigma^0_n$ unsound.

There are two ways to read the above statement. Some people would define the arithmetical hierarchy so that each formula has infinitely many rankings. But even if we only give a formula its lowest ranking, every formula $\phi$ will be logically equivalent to a formula $\psi$ that is $\Sigma^0_n$ for some $n$; just add dummy quantifiers to the prenex form of $\phi$ until it starts with an existential quantifier. Then any theory that proves $\phi$ also proves $\psi$. This is why, for most purposes, we can look at the arithmetical hierarchy "modulo logical equivalence" or even modulo provable equivalence in our theory.