I am aware of Godel, thanks, and of course that choice / Zorn's / well ordering (not to mention CH) are independent of ZF (and in particular that's why I mentio

I am aware of Godel, thanks, and of course that choice / Zorn's / well ordering (not to mention CH) are independent of ZF (and in particular that's why I mentioned ZFC). I'm also willing to accept the existence of an inaccessible cardinal, which bears on ZFC's consistency. Ultimately my belief is this: there was a very real crisis in foundations at the end of the 19th century and over the next several decades this was fixed as best as it could be. The edges of the foundation are not perfect, the edges of the foundation cannot be perfect, but the edges of the foundation have been pushed back so far that for nearly every working mathematician they're good enough. (and if the algebraic geometers need Grothendieck universes, I'm ok with that)