It is my understanding that Gödel's second incompleteness theorem says roughly that
there exists a sentence $\varphi$ such that neither $\varphi$ nor $\neg\varphi$ is provable in Peano's arithmetic. However one of them is true (in the structure $\mathbb{N}$).
If $\varphi$ is a formula in the style : "for all integers $n$, then $\psi(n)$". For me, if $\varphi$ is false, proving it false is simply exhibing a counterexemple $n_f$ such that $\neg\psi(n_f)$. Therefore if it is false, then it must provably false, doesn't it?
Therefore, it someone manage to prove that it not provable, does it mean that $\varphi$ must be true?
Is this reasoning correct?