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There is a question on stack overflow showing that if PA is consistent then PA+¬Con(PA) is also consistent. The thing is, that it should also work vice versa, that if PA+¬Con(PA) is consistent, then PA should also be consistent, but I cфnnot understand how that works or how it could be prooven.

GnKonkort
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    If $T+\varphi$ is consistent, then in particular, $T$ is consistent. You can't prove a contradiction with less axioms if you couldn't prove a contradiction before. – Asaf Karagila Jan 09 '23 at 13:37
  • @Asaf Karagila oh. Yeah sorry, it was just this straightforward. I was trying to overengineer the proof, sorry. – GnKonkort Jan 09 '23 at 13:39

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