5 ms·
I’m a category theorist so I’m going to nitpick - the claim that all concrete mathematical objects are specific sets is either tautological (a concrete category
by eastWestMath 9y ago
I’m a category theorist so I’m going to nitpick - the claim that all concrete mathematical objects are specific sets is either tautological (a concrete category is literally one where each object has an underlying set) or untrue (I can think of several non-concrete categories that may be studied “concretely” using type theory, such as the microlinear spaces of synthetic differential geometry).
- danharaj 9y agoNon-commutative spaces are definitely not sets as well. That might have implications for foundations in the next century.
- jimhefferon 9y agoI'd be interested to hear more. Any reference come to mind?
- jesuslop 9y agoI'm liking https://arxiv.org/abs/math/0408416v1 https://arxiv.org/abs/math/0408416v1
- Verdex_3 9y agoYeah, I would have thought that results like: https://www.scottaaronson.com/blog/?p=2725 https://www.scottaaronson.com/blog/?p=2725 show that set theory isn't the end all be all of mathematics.
- ultrafilter 9y agoEvery 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 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.
- xamuel 9y agoYou'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".
- ultrafilter 9y agoYou're right. I should have written "halts last among its peers" or similar.