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.

  • $\begingroup$ Well, set theorists study other sets of axioms besides ZFC (though they do study ZFC as well), so if you count that, then them. $\endgroup$ – Hayden Feb 26 '15 at 19:15
  • $\begingroup$ Well yes, but that's somewhat trivial $\endgroup$ – Nethesis Feb 26 '15 at 19:15
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    $\begingroup$ Fair enough, but it does show existence! ;) $\endgroup$ – Hayden Feb 26 '15 at 19:16

I think the question suggests a few misunderstandings. A few points, in no particular order:

  • 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.

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  • $\begingroup$ +1. I'd like to make a remark related to what the OP said about needing an extension of ZFC (one with Grothendieck universes) in order to formulate category theory (or more precisely, the theory of large categories). To found category theory on set theory, sure, you have to do this. But we don't have to do that; we can found everything on category theory instead, for instance, among other options like homotopy type theory. $\endgroup$ – Ian Feb 26 '15 at 20:25
  • $\begingroup$ (Cont.) Or we can avoid foundational issues entirely, taking it for granted that other people have already shown that the foundational issues are as resolved as Godel's theorems permit. This is the viewpoint of most mathematicians. $\endgroup$ – Ian Feb 26 '15 at 20:27
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    $\begingroup$ @Ian, i have a couple of issues with what you're saying. Firstly, I don't think there have been any successful attempts to found mathematics on categories. There have been somewhat successful attempts based on sets, and somewhat successful attempts based on $\infty$-groupoids (this is new and exciting, of course), but as far as I know, that's it. Lawvere tried to refound mathematics on categories, but I don't think it really worked, perhaps because he ignored natural transformations. Secondly, how do you know that if you found mathematics on categories, you don't get universes? $\endgroup$ – goblin GONE Feb 28 '15 at 10:17
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    $\begingroup$ @Ian: "This is the viewpoint of most mathematicians." can you back that up, or did you mean "most mathematicians which I know, which is merely a fraction of the entire collection of mathematicians active nowadays in the world; and it might be the case that this opinion is local to my current department but is not as wildly accepted elsewhere." Because if you meant that really most mathematicians think that, I'd be very happy to see some hard evidence for that. In my experience most mathematicians don't care at all about these things. $\endgroup$ – Asaf Karagila Feb 28 '15 at 10:23
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    $\begingroup$ @AsafKaragila I wasn't clear enough. I meant "avoiding foundational issues entirely is the viewpoint of most mathematicians". That statement I would actually stand by as stated. In my experience, though, those that are familiar in passing with some foundational isues tend to wave it off by saying something like "the logicians have already done it as best they can". $\endgroup$ – Ian Feb 28 '15 at 13:52

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