I used large cardinal axioms as [the canonical] example of any axiom stronger than standard mathematical foundations. A contrived example: there are certain la

I used large cardinal axioms as [the canonical] example of any axiom stronger than standard mathematical foundations. A contrived example: there are certain large cardinal axioms that imply the consistency of ZFC. Thus ZFC cannot prove those axioms unless ZFC is inconsistent (Godel's incompleteness theorem). Consider the following problem: "If ZFC can prove 1=0 in n steps, output 1. Else, output 0." A naive solution would brute-force search all ZFC-proofs of length n. A faster solution would be: "Ignore n and immediately output 0." This is correct, because ZFC never proves 1=0. You could formally verify the correctness by using certain large cardinal axioms, but not using raw ZFC.