Examples for intensional set theories.

Question

The normal set theory of todays mathematics (ZFC) is extensional, i.e. it has the axiom of extensionality $$\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow A = B)$$ Are there examples of set theories which do not have an extensional concept of sets, i.e. where the above axiom is not true?

Answer 1

Read the following papers:

P.C. Gilmore, "The consistency of partial set theory without extensionality", Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part 2, American Mathematical Society, Providence, Rhode Island, 1974, pp. 147-153.

M. Forti and R. Hinnion, "The Consistency Problem For Positive Comprehension Principles", Journal of Symbolic Logic, Vol. 54, No. 4 (Dec., 1989), pp. 1401-1418.

R. Hinnion, "About The Coexistence of 'Classical Sets' with 'Non-Classical' Ones: A Survey", Logic and Logical Philosophy, Vol 11, (2003), pp. 79-90.

O. Esser, "On the axiom of extensionality in positive set theory", Mathematical Logic Quarterly, 49, No. 1, pp 97-100.

I quote from the third paper, from section 2, "Partial Sets":

"This kind of concept was introduced by Gilmore [in the first paper listed here--my comment] who proposed two versions (PST and $PST^{+}$) of a theory of 'partial sets'...Those theories negate the corresponding axiom of extensionality...."