The OP goes on to immediate ask whether category theory or type theory can serve as a foundation for mathematics (they give a link to an article that talks of "

The OP goes on to immediate ask whether category theory or type theory can serve as a foundation for mathematics (they give a link to an article that talks of "foundational styles" but which doesn't give a process of explicitly "cutting the cord" from set theory). I think the answer by (perhaps infamous) Paul Taylor in this mathoverflow post[1] is a clearer expression of the positions of those who want to jettison traditional set theory as the foundation of mathematics. "The crucial issue is whether you are taken in by the Great Set-Theoretic Swindle that mathematics depends on collections (completed infinities)" Yes, it's important to remember that everything labeled "infinity" in math doesn't "exist-exist" but rather is some combination of unending process and arbitrary labeling (see Skolem's paradox etc). But that doesn't mean such nominally completed infinities aren't useful or even necessary for a human being to grasp mathematics. Anyway, for this debate it's worth reading all of Taylor's comment (which isn't to say I share his position but it's good to understand where the "finitests" are coming from). [1] https://mathoverflow.net/questions/8731/categorical-foundations-without-set-theory https://mathoverflow.net/questions/8731/categorical-foundati...