It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
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As soon as Gödel published his first incompleteness theorem, I would have thought the entire field of mathematics would have gone full throttle on trying to find more axioms. Instead, over the almost century since then, Gödel’s work has been treated more as an odd fact largely confined to niche foundational studies rather than any sort of mainstream program (I’m aware of Feferman, Friedman, etc., but my point is there is significantly less research in this area compared to most other topics in mathematics).
But why? Gödel's theorem does not depend on number of axioms but on them being recursively enumerable.
If you’re talking about every true sentence in the language of PA, then not all such sentences are derivable via the theory of PA. If you are talking about the theorems of PA, then these are missing an infinite number of true statements in the language of PA.
Harvey Friedman’s “grand conjecture” is that virtually every theorem that working mathematicians actually publish can already be proved in Elementary Function Arithmetic (much weaker than PA in fact). So the majority of mathematicians are not pushing the boundaries of the existing foundational theories of mathematics, although there is certainly plenty of activity regardless.