# Book:Paul J. Cohen/Set Theory and the Continuum Hypothesis

## Paul J. Cohen: Set Theory and the Continuum Hypothesis

Published $1966$, Benjamin.

### Contents

Preface
Chapter I. General Background in Logic
1. Introduction
2. Formal Languages
3. Universally Valid Statements
4. Gödel Completeness Theorem
5. The Löwenheim-Skolem Theorem
6. Examples of Formal Systems
7. Primitive Recursive Functions
8. General Recursive Functions
9. Gödel Incompleteness Theorem
10. Generalized Incompleteness Theorem
11. Further Results in Recursive Functions
Chapter II. Zermelo-Fraenkel Set Theory
1. Axioms
2. Discussion of the Axioms
3. Ordinal Numbers
4. Cardinal Numbers
5. The Axiom of Regularity
6. The System of Gödel-Bernays
7. Higher Axioms and Models for Set Theory
8. Löwenheim-Skolem Theorem Revisited
Chapter III. The Consistency of the Continuum Hypothesis and the Axiom of Choice
1. Introduction
2. Proof of Theorem 1
3. Absoluteness
4. Proof of AC and GCH in L
5. Relations with GB
6. The Minimal Model
Chapter IV. The Independence of the Continuum Hypothesis and the Axiom of Choice
1. Introduction
2. Intuitive Motivation
3. The Forcing Concept
4. The Main Lemmas
5. The Definability of Forcing
6. The Model N
7. The General Forcing Concept
8. The Continuum Hypothesis
9. The Axiom of Choice
10. Changing Cardinalities
11. Avoiding SM
12. GCH Implies AC
13. Conclusion
References