Set Theory

Set theory is concerned with the concept of a set, essentially a collection of objects that we call elements. Because of its generality, set theory forms the foundation of nearly every other part of mathematics.

Before you begin

In order to make things easier for you as a reader, as well as for the writers, you will be expected to be familiar with a few topics before beginning. (I hope to have some links to other Wikibooks here soon.)

• Mathematical Logic & Proofs
  • Mathematics is all about proofs. One of the goals of this book is to improve your skills at making proofs, but you will not learn any of the basics here.
  • Many constructions in set theory are simply generalizations of constructions in mathematical logic, and therefore logic is a necessity of learning set theory.

Set theory

Introduction

1. Sets
2. Axioms
3. Relations
4. Orderings
5. Zorn's Lemma and the Axiom of Choice
6. Ordinals
7. Cardinals
8. Zermelo-Fraenkel Axiomatic Set Theory

Appendix 1. Naive Set Theory

Review

Further reading

• Discrete Mathematics/Set theory
• Krzysztof Ciesielski, Set Theory for the Working Mathematician (1997)
• P. R. Halmos, Naive Set Theory (1974)
• Karel Hrbacek, Thomas J. Jech, Introduction to set theory (1999)
• Thomas J. Jech, Set Theory 3rd Edition (2006)
• Kenneth Kunen, Set Theory: an introduction to independence proofs (1980)
• Judith Roitman, Introduction to Modern Set Theory (1990)
• John H. Conway, Richard Guy The Book of Numbers - chapter 10
• Tobias Dantzig, Joseph Mazur Number: The Language of Science