Category theory? Logic? Anyone experienced this like me?

Question

Mathematics is not logic, but if one uses Zorn's lemma and stuff he should accept logical impact on mathematics. I'm the one who cares a lot about logics.

It seems like Category theory is inevitable in every branch of mathematics at some level. However, category theory is kind of different logical framework than the standard set theory ZFC. I read articles about this paradoxical situation over and over, but still I am not satisfied. It seems like people are switching their logical framework from one to another whenever they need

I found that the only known way to do Category theory in ZFC is to accept "Grothendieck's universe" which is equivalent to the existence of strongly inaccessible cardinal.

I think this is really painful since inaccesible cardinal axiom proves the consistency of ZFC.

Is there anyone who had been suffered for accepting Category theory? What was your solution?

Answer 1

I think there are reasons to have a pragmatic attitude to sets in category theory (and perhaps in all mathematics). I like the way that Adamek-Herrlich-Strecker present this topic in Abstract and Concrete Categories p. 13-16:

Someone working, for example, in algebra, topology, or computer science usually isn’t (and needn’t be) bothered with such set-theoretical difficulties. But it is essential that those who work in category theory be able to deal with “collections” like those mentioned above. To do so requires some foundational restrictions. Nevertheless, certain naturally arising categorical constructions should not be outlawed simply because of the foundational safeguards. Hence, what is needed is a foundation that, on the one hand, is sufficiently flexible so as not to unduly inhibit categorical inquiry and, on the other hand, is sufficiently rigid to give reasonable assurance that the resulting theory is consistent, i.e., does not lead to contradictions.

For most mathematicians the big deal with axiomatic set theory is that it proves that sets exists and that it give rules how sets can be created.

In ACC the hierarchy sets-classes-conglomerates is used to present category theory.