Non-Well-Founded Set Theory

Abstract

The approach to set theory that has motivated and dominated the study presented so far in this book has essentially been one of synthesis: from an initial set of axioms, we build a framework of sets that can be used to provide a foundation for all of mathematics. By starting with pure sets provided by the Zermelo—Fraenkel axioms, and progressively adding more and more structure, we may obtain all of the usual structures of mathematics. And then, of course, we may make use of those mathematical structures to model various aspects of the world we live in. In this way, set theory may be used to provide ways to model ‘mathematical’ aspects of our world.