When I took, last year, my university's course on set theory, I remember my teacher introduced the topic by saying that ZFC Set Theory is an axiomatic system over which math is founded, but it's not the only one since there are many other of these, that, just like ZFC, are built upon first order logic. But then he said one can go further and find other axiomatic systems that serve as a foundation, built upon other logics or theories; He cited topoi as the axiomatic system built upon category theory that serves as a foundation for math. Then he mentioned there are analogous axiom systems built upon fuzzy logic, but said there are not such systems built upon modal logic.
I would like to know
- Is it true that there are not axiomatic systems built upon Modal Logic that serve as a foundation of mathematics? If so, why is that? Where can I find more info. about this?
- Where can I find more information on all the axiomatic systems there are, and the logics and theories they are based upon? What is this area of math called?