Dear Math Friends on G+:
This is unsettling and I am too embarrassed to post this on a Math forum like stack exchange or Mathoverflow.
I have been worried about the foundations of Category theory as needed for studying Algebraic Geometry and its applications to Number theory (yes, I do have things like Etale Cohomology in mind). I don't believe that universes as envisioned by Grothendieck and subsequently developed in detail by several people form the right foundation for this. This is because it leads to inaccessible cardinals and one can demonstrate that "their existence is consistent with ZFC" cannot be proven from ZFC.
However, it appears to me that from the point of view of the applications I have in mind that I must admit "some" proper classes in the universe of discourse along with ZFC. Is this right? Or is it true that one can work inside ZFC always (and deal only with small categories)?
(This may be naive, but someone can set me straight on this while we are at it: is it true that admitting a proper class amounts to admitting a strongly inaccessible cardinal?)
Thank you in advance.
In addition, I am ccing this to +David Roberts
, Prof. +Chandan Dalawat
, Prof. +Pierre-Yves Gaillard
, Prof. +Amritanshu Prasad
who all might have something to say about this. (Sorry if this comes across as a rude post.)