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Symmetry as a criterion for sethood, motivating Quine's set theory NF

### M. Randall Holmes

The idea that "smallness" in a suitable sense is what determines whether a class is a set is well known, motivating the von Neumann-Godel-Bernays and Morse-Kelley theories of sets and classes, which are alternative presentations of the usual Zermelo-style theory of sets.

In this talk, we will present a theory of sets and classes in which the criterion for sethood is **symmetry** in a suitable sense: the sets in this theory turn out to satisfy the axioms of Quine's set theory New Foundations.

This is interesting in part because a usual criticism of Quine's NF is that its motivation is purely syntactical: this gives a possible semantically based motivation for the theory.

We are not attempting here to prove that the theory is consistent (and in fact we do not know how to prove the consistency of the theory of sets and classes we describe).