Notes on Set Theory

The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. It is also viewed as a foundation of mathematics so that "to make a notion precise" simply means "to define it in set theory." This book gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation.

Contenu

Introduction Equinumerosity Paradoxes and axioms Are sets all there is? The natural numbers Fixed points Well ordered sets Choices Choice's consequences Baire space Replacement and other axioms Ordinal numbers A. The real numbers B. Axioms and universes Index