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Yeah, I would have thought that results like: https://www.scottaaronson.com/blog/?p=2725 show that set theory isn't the end all be all of mathematics.

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Every formal axiom system, whether about sets or anything else, can only prove that N of Busy Beavers halt, for some (not-too-large in practice) finite number N. If you just want to maximize N, right now your best bet is set theory plus a very strong large cardinal axiom, like I0.

http://cantorsattic.info/L_of_V_lambda%2B1

Perhaps simply because it is older, set theory is way ahead of its competitors in developing a hierarchy of extremely strong (but apparently consistent) axioms to supplement its base theory ZFC.

You're confused. Only very basic arithmetic is needed to prove that any particular TM halts (if it does halt). You might have meant to say something like, "can only prove that N of the Busy Beavers are really Busy Beavers".

You're right. I should have written "halts last among its peers" or similar.

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