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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.

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Because "the rest of mathematics" is much larger than "the rest of the mathematics we know and care about." –  Qiaochu Yuan Feb 14 '11 at 23:37
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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. –  Matt Feb 15 '11 at 0:39
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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. –  Arturo Magidin Feb 15 '11 at 3:14
    
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. –  Stefan Walter Feb 16 '11 at 13:37
    
Tom Leinster speaks about this issue in arxiv.org/abs/1012.5647 in some depth. –  Thomas Rot Feb 18 '11 at 12:34
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2 Answers

There are two ways to answer your question:

  • what does foundation mean? The two areas address 'foundation' from different perspectives. Set theory (and logic) are attempts at foundations of mathematical knowledge about how we can know mathematical truths - how do you know that a theorem is 'true' (and what does true mean) ? (set theory does this by allowing a translation from your favorite branch of mathematics to sets, and then formalizing the inference operations on just the set objects; everything can be converted to a set)

Category theory is a foundations for -what- we know about mathematics - what similarities/commonalities are among disparate areas of mathematics. Take your favorite brach of mathematics, define some objects and some suitable functions and then you get all sorts of theorems for free (everything can be viewed through the 'category' lens. everything is a category)

Of course you can use one of these for the other (give a -logical- foundation of category theory, e.g. the objects and morphisms are implemented with sets; give a -categorical- foundation for set theory/logic, e.g. one can have a category of 'set' with 'functions' as the morphisms).

  • as to your title, how can there be more than one foundation at all, even of the same kind?There can certainly be more than one truth foundation (intuitionistic, constructivist, allowing different set axioms, different rules of inference, ZFC, NBG, etc.) each allowing desired features and disallowing unwanted features. And there have been many unification foundations (set theory can be viewed as one (just sets as the things that can be manipulated), group theory, algebra as a whole (symbolic manipulation), just category theory is the latest and best unification.
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I'm learning about some of this right now! As far as I know, you already need quite a lot of set theory to say topos. An (elementary) topos is is a category closed under a lot of common constructions.

These topos gadgets also have an internal logic (usually intuisionistic), and given for instance a natural number object you can perform many of the well-known constructions of mathematics, such as those of the real numbers (here the Cauchy and Dedekind reals may be different). This is what we do in the topos of sets!

An elementary topos $E$, by the way, is a cartesian closed category (which means it has a terminal object $1$, products, exponentiation, and an adjunction between the two, i.e. $\hom_E(X\times Y,Z)\cong\hom_E(X,Z^Y)$) with a subobject classifier $\Omega$ together with a morphism $\top:1\to\Omega$ called true, such that any monic $f:Y\to X$ is the pullback of a unique morphism $\chi(f):Y\to\Omega$. We also require pullbacks along $\top$ to exist. It follows that $E$ has finite limits and colimits, and that the subobject functor $Sub:E\to Set$ is represented by $\Omega$ (i.e. $Sub(-)\cong\hom(-,\Omega)$).

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Not true. The theory of elementary toposes is first-order, as is the theory of categories. (The theory of Grothendieck toposes, on the other hand, is not, as far as I know.) –  Zhen Lin Apr 9 '11 at 11:32
    
You talking about the appendix which I removed? Sheaf topoi are Grothendieck topoi. I'm not entirely sure how this all ties together. But it's a huge and interesting field no less. –  Eivind Dahl Apr 9 '11 at 11:59
    
I was referring to your comment "As far as I know, you already need quite a lot of set theory to say topos.", my point being that we can obviate the need for set theory by going to a first-order axiomatisation. –  Zhen Lin Apr 9 '11 at 14:09
    
Interesting! I've been hoping you could get around it somehow. –  Eivind Dahl Apr 9 '11 at 14:44
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