[red_trumpet]

I always find it a bit weird, when people compare set theory to category theory like this. When talking about set theory as a foundation for mathematics, I always think of Zermelo-Fraenkel[1] set theory (possibly with Choice), which is an axiomatic system, from which you can build a lot of maths (at some point one might want to introduce universes[2], but whatever). I'm not aware of a similar axiomatic system using category theory, are you?

[1] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... [2] https://en.wikipedia.org/wiki/Grothendieck_universe

[tupshin]

I've found the best layman's grounding for Category theory's relationship to axioms to be in Bartosz's post here:

https://www.quora.com/Are-there-any-axioms-for-category-theo...

[red_trumpet]

That's a nice post, and it actually reinforces my thoughts: You already need a foundational theory before coming to category theory.

[sylware]

Well... then set theory with core logic still THE foundation then.

[n4r9]

Yeah, I deliberately avoided using the word "foundation" as I wasn't too confident about that. But I believe it makes sense to say that category theory and set theory both seek to formalise and generalise much of mathematics? Even if the first is more of a sort of framework than a foundation.