[EtDybNuvCu]

Pick the Grothendieck-Tarski axiom instead, and use category theory to build ZFC via topos. This path is "big" enough to handle all the interesting sets; it can't deal with proper classes, but proper classes are kind of metaphysical anyway.

[0] https://en.wikipedia.org/wiki/Tarski–Grothendieck_set_theory

[ginnungagap]

Sure, but ZFC by itself also deals with every interesting set, I was just being nitpicky of my own assertion.

I'm not familiar with TG, what's the relation between it and ZFC+some large cardinal axiom?