Universes of Sets

Abstract

The credo of naive set theory includes, as is known, the dream about the ‘Cantor paradise’, i.e., about the universe, the world of sets, containing all possible formations that can be thought of as distinct. Realistic approximations to the unattainable ideal, i.e., adequate formal schemes making it possible to present a large spectrum of concrete sets while remaining within comfortable conditions of sufficient logical accuracy, are the subject of the modern set theory. In the first part of the book we have already encountered the ideas underlying a number of axiomatic set theories, such as Zermelo-Fraenkel set theory, the theories of external and internal sets.