Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?
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asked Jan 20, 2019 at 12:26
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4$\begingroup$ If you formulate everything correctly, ECTS + separation is equivalent to ZFC. There is a translation you need to do because ECTS is two-sorted. See Tom Leinster's nice summary article: arxiv.org/abs/1212.6543 $\endgroup$– Harry GindiCommented Jan 20, 2019 at 12:33
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$\begingroup$ Thanks for the reference. $\endgroup$– Jörg NeunhäusererCommented Jan 20, 2019 at 12:48
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1$\begingroup$ I think I meant replacement, not separation. It's whatever Leinster calls axiom 11. $\endgroup$– Harry GindiCommented Jan 20, 2019 at 13:19
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1 Answer
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Yes, this is a well-known fact that goes back to Cole, Mitchell, and Osius in the 70's. The relevant kind of category is, as Harry says in the comments, a well-pointed topos with extra properties; Lawvere's Elementary Theory of the Category of Sets is an axiomatization of such a category. To define ZF-sets in such a category you build them along with their hereditary membership structure as well-founded graphs; there is a good introduction in Chapter VI of Mac Lane & Moerdijk's book Sheaves in Geometry and Logic. If you want a very detailed treatment that includes analogous results for a wide variety of set theories including both ZF and ZFC and also weaker and constructive versions, there is my own paper Comparing material and structural set theories.
answered Jan 20, 2019 at 15:31
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$\begingroup$ You linked the v2 not the v3 of your paper. Was that intentional $\endgroup$ Commented Jan 20, 2019 at 15:35
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$\begingroup$ Thanks, i will use the references. $\endgroup$ Commented Jan 20, 2019 at 16:47
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2$\begingroup$ @HarryGindi Yes, the two versions are identical, but v2 has the more permissive license. (-:O $\endgroup$ Commented Jan 20, 2019 at 16:53