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I have always assumed that extensionality is a paradigmatic example of a property of mathematical objects (sets) which is essential to those objects--- if your set theory doesn't obey extensionality, it isn't set theory.

Given the existence of alternative set theories, such as non-well-founded set theories, though, it occurred to me that maybe I shouldn't be so certain of this.

Which brings me to my question: have any set theories been developed which don't include the axiom of extensionality, or at least which include set-like objects which are not extensional?

This question is certainly related to my own, but doesn't consider theories which have been developed, only ways that extensionality might fail by manipulating models of ZFC.

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  • $\begingroup$ Set theory with atoms is technically not extensional, but that's just a problem with the definition. $\endgroup$ – Zhen Lin Nov 1 '13 at 18:15
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    $\begingroup$ See also here. $\endgroup$ – Andrés E. Caicedo Nov 1 '13 at 19:24
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Various kinds of type theories are intensional by nature, and in those theories one can usually carry out the constructions of typical set theory, and therefore deal with sets in an intensional way.

I suggest that you look at the book Intensional Mathematics, which is part of the series Studies in Logic and the Foundations of Mathematics published by Elsevier. In it, you will find two articles, one by J. Myhill called Intensional Set Theory, and one by N. D. Goodman called A Genuinely Intensional Set Theory (which is kind of a follow-up to Myhill's article). They develop intensional modal set theories, and Goodman also gives an interpretation of ZFC in ZFM (which is his proposal of a modal Zermelo-Fraenkel).

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