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How can set theory and category theory both be plausible theories for the foundations of mathematics? If these two theories are not mathematically equivalent, does it not mean that the rest of mathematics, when taken exclusively with either of these two theories as foundation, will be distinct; that is, unless these two theories are, in some way, equivalent.

Thanks

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    Because "the rest of mathematics" is much larger than "the rest of the mathematics we know and care about."2011-02-14
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    I haven't ever really looked into category theory as a foundation, but I've always assumed it contains set theory foundations in some sense.2011-02-15
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    @Matt: Not really; your primitive notions are things like "object", "arrow", etc. You do have a notion of "function" that precedes the theory (to talk about the domain and codomain assignment) but this is to some extent also true in Set Theory, with the Axiom of Replacement.2011-02-15
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    Please correct me if I'm wrong: I have the impression that the role of category theory in foundations is not to replace set theory, but rather to provide a framework for studying different set theories.2011-02-16
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    Tom Leinster speaks about this issue in http://arxiv.org/abs/1012.5647 in some depth.2011-02-18
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    actually set theory is a special case of category theory, and can be derived from it. You can read more about category theory from Topoi by Goldblatt.2011-04-09
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    @Matt Barry Mazur has a very nice essay that discusses the sense in which category theory does, and does not, depend on set theory: [When is one thing equal to some other thing?](http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf)2013-06-26

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