Non-standard axioms + ZF and rest of math I've never taken a formal course of Set Theory, but I've been wondering about this for some time now.
Are non-standard axioms, like $\mathbb{V}=\mathbb{L}$ and axioms about large cardinals and any other you can think of (which are independent of $ZFC$) used outside pure Set Theory?
Are there some non-trivial problems in other branches of math (like algebra, topology, analysis etc.) that can only be solved if these axioms are used? Can they permit pathological behaviour in those fields?
If not, why don't we choose the axioms to make sets behave in the nicest way possible like $\mathbb{V}=\mathbb{L}$, and use them to simplify things in the Set Theory itself? Why is $ZFC$ so universally used?
Also, are there still some branches of math, that reject axiom of choice? 

Grothendieck universes, developed by Grothendieck for use in algebraic geometry, and which have applications in category theory, are equivalent to strongly inaccessible cardinals.
