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One knows that many models of set theory exist. In topos theory,"the" category of sets is to play the role of the point. Since many models of set theory are around, I believe one of the following to be true.

  1. There is one category of sets and the model determines what is true about the category of sets.

  2. There are many different categories that can be called the category of sets, one for each model.

The question is: is 1 or 2 correct? Perhaps neither one is correct. In either case,what would a good reference be?

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Here is a thoroughly Platonist answer to your question: Both 1 and 2 are true. There is one category of sets. Its objects are all of the sets, and its morphisms are all of the functions between them. But there are many other categories that can be (and in fact have been) called the category of sets (by abuse of language). In fact, I often think of an arbitrary elementary topos as the category of sets.

Most non-Platonist mathematicians would say that 1 is false and 2 is true, or perhaps that one shouldn't speak of the category of sets at all but rather of a category of sets.

A few mathematicians, who don't believe the consistency of any sort of set theory, might say that there are no categories that can be called the category of sets because there are no models of set theory.

In an effort to clarify these positions, their respective advocates would assert them more loudly.

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Just to playfully quibble, some platonist might think that 2 is true if their is a muliverse. –  Lunasaurus Rex Sep 29 '11 at 21:02
    
+1 For the last line. Hopefully we can avoid this trap. –  BSteinhurst Sep 29 '11 at 22:41
    
I don't get the point of that last line. Is it to stifle dissent? Is it provocation? Is it helpful? –  Todd Trimble Jul 28 at 21:06
    
@ToddTrimble The last line was intended to be a somewhat cynical, somewhat humorous observation on the "discussions" I've seen; usually people are just talking past each other. I hope that it would stifle, not all dissent, but the sort of dissent and also the sort of agreement that it describes. I have to admit, though, that for almost 3 years, it seems to have stifled everything except your comment. –  Andreas Blass Aug 1 at 17:39
    
It's possible that by "advocates", you meant non-Platonists and Platonists alike. I at first read it as referring to advocates of non-Platonist positions as being 'loud' (I thought elsewhere I'd seen you more or less openly declare your Platonist sympathies, which I'd guess makes you one of the majority). I agree that many such discussions are 'loud' (metaphorically or actually), but I wouldn't say any one group is louder than another. –  Todd Trimble Aug 4 at 16:39

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