basic train of thoughts: Gödel's incompleteness theorem (and Turing's subsequent adaptation to computational machinery) shows that mathematics itself is incomplete. that was a shock: mathematics is the domain of all formal languages, but mathematics cannot be generated using them
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That depends on the axiom and the logic, as an example again, what the culture of mathematics have decided to call the axiom of choice is provable within intuitionistic dependent type theory.
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The axiom of the excluded middle isn't actually a stated or independent axiom in all natural deduction formulations of classical logic either. i.e. its provable within them but not one of their axioms, yet we agree that it is an axiom
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