Con(PA) implies consistency of $\mathsf{PA}$ + ¬Con($\mathsf{PA}$) The Wikipedia article for $\omega$-consistency says "Now, assuming PA is really consistent, it follows that $\mathsf{PA}$ + ¬Con($\mathsf{PA}$) is also consistent, for if it were not, then PA would prove Con(PA) (since an inconsistent theory proves every sentence), contradicting Gödel's second incompleteness theorem." I'm not sure how that follows; if $\mathsf{PA}$ + ¬Con($\mathsf{PA}$) is inconsistent, then it can obviously prove Con($\mathsf{PA}$), but I don't get how that shows that $\mathsf{PA}$ could prove Con($\mathsf{PA}$).

If $\sf PA+\lnot \rm Con\sf (PA)$ is inconsistent but $\sf PA$ is consistent, then in every model of $\sf PA$ it is true that $\rm Con\sf (PA)$, now by completeness we get that Peano proves its own consistency.