Which areas of modern mathematics don't use ZFC set theory? I've heard that Algebraic Geometry requires something called Category Theory, which itself requires an extension of ZFC called Tarski-Grothendieck set theory, and that got me wondering.
Which areas of mathematics today do not use ZFC as their axioms? By "use ZFC" I mean entirely use prior results which use prior results which .... which are logically derived from ZFC. 
 A: I think the question suggests a few misunderstandings. A few points, in no particular order:


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*No area of mathematics is based on explicitly given axioms, possibly with the exception of set theory and/or certain branches of Euclidean geometry; unfortunately, that is just not how fields of mathematics are structured. Certain very rigorous books are structured like this, but a book is not an entire field.

*You cannot trace results all the way back to the ZFC axioms, because mathematics is (currently) more like a network of results than a tree, and it is full of loops, and even, to some extent, gaps.

*Realistically, category theory is so widespread today that if you want to formalize things within a ZFC-style set theory, you're going to have to use axioms that go beyond the usual ZFC axioms and postulate the existence of universes.

*Arguably, the whole point of ZFC and related systems is to have their limitations studied. For example, it has been known for a long time now that ZFC can neither prove nor refute the continuum hypothesis; obviously, you have to formalize what you mean by "ZFC" very carefully to be able to prove something like this. Nobody founds mathematics on ZFC "in practice"; rather, set theorists develop enough math within ZFC to show that it can be done "in principle" and to get an intuition for what's possible and what's not, and then go on to study the limitations of ZFC and its various extensions.

*People who are interested in founding mathematics in practice (not just in principle) usually have one foot in the mathematics camp and one foot in the computer science camp, and they usually study type theory.
