I don't know much about logic, but similar to how certain theorems "require" their proofs to use the axiom of choice, are there theorems which "require" induction? Is there a way to detect them?
Thanks!
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I suspect -- although you could correct me if I'm wrong -- that your question isn't really the same as the one it was marked as a duplicate of. At least the answers there seem to alternately explain (1) the idea of necessity only for Peano arithmetic, or (2) more practical questions of how to find some proofs which are non-inductive when the most well-known proof is inductive. I think what you're asking is more like "Not just for Peano arithmetic but for all of the consequences of ZFC, which of them require induction and which do not?" – Addem Jul 16 '23 at 20:51
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If I am right, that the duplicate doesn't really address your question, then I think the best answer is: Induction isn't an axiom of ZFC, so it may not even make sense to talk about theorems which require it. Induction is a consequence of ZFC, and certainly we could try to understand which axioms are necessary in the proof of the validity of induction. And then you could ask necessity questions about those axioms. – Addem Jul 16 '23 at 20:54