> It's a rather big philosophical stance to say that the 'truth' of these propositions can be determined outside of the axiomatic systems.
I don’t think so. ZFC is either consistent or it isn’t. That is, a Turing machine enumerating all theorems of ZFC either produces 0=1 or it doesn’t. This is, however, independent of ZFC (if the latter is consistent) by the second incompleteness theorem. See https://mathoverflow.net/a/332266, in particular Nik Weaver’s comment:
> I think the strongest argument in favor of the definiteness of ℕ, and against the idea that PA, or any other axiomatization, is the be-all end-all, is the evident fact that we have an open-ended ability to go beyond any computable set of axioms, for instance by affirming their consistency. If you accept PA you should accept Con(PA), and the process doesn't stop there: you can then accept Con(PA + Con(PA)), and so on. This goes on to transfinite levels. If our understanding of ℕ really were fully captured by some particular set of axioms then we would not feel we had a right to strengthen those axioms in any special way; the fact that we do feel we have this right shows that our understanding is not captured by any particular set of axioms.
I'm not 100% sure this perspective works, but could one take the position that this is because of a disconnect between model and reality?
You look at some big list of symbols and say "this formula (model) corresponds to whether a Turing machine enumerating all theorems of ZFC produces 0=1 (reality)". But does it really?
In a way. Consider a Turing machine that enumerates all proofs in ZFC and halts iff it finds a proof of 0=1 (i.e. a contradiction). The statement that this Turing machine halts (like the statement that any given Turing machine halts) is what’s called a Sigma_1 statement, which lies at the bottom of the arithmetic hierarchy (right above bounded-quantifier statements). ZFC + ~Con(ZFC) is consistent but Sigma_1-unsound and therefore omega-inconsistent [1].
I don’t think so. ZFC is either consistent or it isn’t. That is, a Turing machine enumerating all theorems of ZFC either produces 0=1 or it doesn’t. This is, however, independent of ZFC (if the latter is consistent) by the second incompleteness theorem. See https://mathoverflow.net/a/332266, in particular Nik Weaver’s comment:
> I think the strongest argument in favor of the definiteness of ℕ, and against the idea that PA, or any other axiomatization, is the be-all end-all, is the evident fact that we have an open-ended ability to go beyond any computable set of axioms, for instance by affirming their consistency. If you accept PA you should accept Con(PA), and the process doesn't stop there: you can then accept Con(PA + Con(PA)), and so on. This goes on to transfinite levels. If our understanding of ℕ really were fully captured by some particular set of axioms then we would not feel we had a right to strengthen those axioms in any special way; the fact that we do feel we have this right shows that our understanding is not captured by any particular set of axioms.