i wrote a mathoverflow answer about godel's incompleteness theorem and nonstandard models of arithmetic that believe arithmetic is inconsistent that some of you might be interested in:
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Some people say that they believe that PA is consistent despite not being able to prove it and I still don't get that. It seems like we can only be in two states of knowledge about it. Either we are uncertain about the consistency of PA, or we know it is inconsistent.
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so if you ask a mathematician who knows a little about this, the standard boilerplate answer is that ZF proves the consistency of PA because you can construct a model of PA in it (the standard natural numbers). of course then you can ask why they believe that ZF is consistent
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(and some mathematicians don't!)
mostly mathematicians just ignore this issue because it doesn't affect how they do math in practice at all. i think most mathematicians are basically platonists and they think N platonically exists and that's a fine way to do things i guess
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also, first-order PA is just one particular formalization of what it means to write down proofs in arithmetic and mathematicians are in no way bound by it; e.g. second-order PA has a stronger induction axiom and no non-standard models, but second-order logic sucks
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It's also really weird in that (when you do it naively??) you are quantifying over things belonging to a different theory, namely some set theory, at least when you do things classically. That always bothered me
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