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I know there are at least three foundations of math:

  1. Zermelo-Frankel set theory. (Sp?)
  2. ECTS
  3. Category theory

Are there any others? Experimental foundations are welcome.

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  • $\begingroup$ Euclid's elements was the first foundations of mathematics. $\endgroup$ – Somos Feb 23 at 21:25
  • $\begingroup$ Re. 1, the spelling is "Zermelo-Fraenkel". Re. 2, it's "ETCS", "Elementary Theory of the Category of Sets". $\endgroup$ – Alexis Feb 24 at 0:24
  • $\begingroup$ If you're interested, you should also take a peek into the underlying logics and philosophies. For example, what happens when you reject excluded middle? What are the connections to category theory and type theory? Or in set theory, there are different approaches, like constructive set theories. All that stuff is pretty related $\endgroup$ – user2103480 Feb 24 at 4:10
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NF(U). We don't know yet, but Randall Holmes claims, that NF is consistent relative to ZFC. NFU is consistent relative to ZFC (and even BZC). But the category of NF sets is not a topos, so it's weird.

There are all sorts of variants of material set theory, some with non-classical logics, some permitting proper classes:

There is also the structural set theory SEAR, devised by Mike Shulman as an example of structural sets not axiomatised by properties of the category they define.

One could also just take a dependent type theory, such as Martin-Löf Type Theory (MLTT, but properly there are historical variants).

CCAF (Category of Categories As Foundation) was proposed by Lawvere, but didn't really take off. See discussion at this MathOverflow question as to how it could be improved.

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  • $\begingroup$ You can find a decent (and working) treatment of CCAF here: McLarty, C., Axiomatizing a category of categories. Journal of Symbolic Logic 56 (1991, no. 4), 1243–1260 (doi:10.2307/2275472, Project Euclid, JSTOR) $\endgroup$ – David Roberts Feb 25 at 11:49
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Univalent Foundations, using Homotopy Type Theory, is another.

(More generally, I like this quote by John Baez: "[M]y opinion is mainly that people should try all sorts of foundations and see what happens.")

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