Property Theory and the Foundations of Mathematics (abstract)

Goedel once said "There never were any set-theoretic paradoxes, but the property-theoretic paradoxes are still unresolved." I will explain this, indicate the intended distinction between sets and properties (which isn't just over extensionality), and review work since Goedel toward resolving the property-theoretic paradoxes. (In my view we now have a pretty good, though not completely satisfactory, solution to them, and there is a reasonable hope of improving it.) I will also discuss the role of properties in the foundations of mathematics, as a replacement of proper classes and to implement an appropriate version of higher order logic.