If I am not wrong, every $\Sigma_n$ (or $\Pi_n$ ) statement $\phi$ is equivalent to a statement that says that a given Turing machine halts (or doesn't halt) on input $C$ using a $\Sigma_{n-1}$-oracle. Then, because we do not have oracles, we shouldn't be able to prove any such statement with $n\geq 2$ within PA, even if $\phi$ is true in the standard N.
If the answer to the question is yes, does that mean that any theory stronger than PA that proves a $\phi$ true in N (but independent of PA) equivalent to introducing axioms about oracles? Are then math theories (or formal systems) stronger than PA ontologically equivalent to educated guesses about oracles?
Bonus question: Is the CH or any other axiom of set theory and beyond equivalent to some kind of transfinite halting question using some kind of "super" oracle?
More distilled question: I agree that even if ϕ is independent from PA, ϕ∨¬ϕ is not, because it is a tautology. Then, for instance, if ϕ∨¬ϕ is Σ3, then ϕ∨¬ϕ is equivalent to state that a given Turing machine halts on input C using a Σ2-oracle (and that will be true). Now, does the fact that we were able to prove ϕ∨¬ϕ mean that the oracle is not necessary? or stating it in a different way, doest it mean that, even if ϕ∨¬ϕ is Σ3, and because it can be proved, it is also equivalent to state that the same Turing machine will halt on input C even without an oracle?