# Book:Thomas J. Jech/The Axiom of Choice

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## Thomas J. Jech:

## Contents

## Thomas J. Jech: *The Axiom of Choice*

Published $\text {1973}$, **Dover Publications**

- ISBN 0-486-46624-8.

### Subject Matter

### Contents

- Preface (December 1972)

- $1$. Introduction
- 1.1. The Axiom of Choice
- 1.2. A nonmeasurable set of real numbers
- 1.3. A paradoxical decomposition of the sphere
- 1.4. Problems
- 1.5. Historical remarks

- $2$. Use of the Axiom of Choice

- 2.1 Equivalents of the Axiom of Choice
- 2.2 Some applications of the Axiom of Choice in mathematics
- 2.3 The Prime Ideal Theorem
- 2.4 The Countable Axiom of Choice
- 2.5 Cardinal numbers
- 2.6 Problems
- 2.7 Historical remarks

- $3$. Consistency of the Axiom of Choice

- 3.1 Axiomatic systems and consistency
- 3.2 Axiomatic set theory
- 3.3 Transitive models of $\text {ZF}$
- 3.4 The constructible universe
- 3.5 Problems
- 3.6 Historical remarks

- $4$. Permutation models

- 4.1 Set theory with atoms
- 4.2 Permutation models
- 4.3 The basic Fraenkel model
- 4.4 The second Fraenkel model
- 4.5 The ordered Mostowski model
- 4.6 Problems
- 4.7 Historical remarks

- $5$. Independence of the Axiom of Choice

- 5.1 Generic models
- 5.2 Symmetric submodels of generic models
- 5.3 The basic Cohen model
- 5.4 The second Cohen model
- 5.5 Independence of the Axiom of Choice from the Ordering Principle
- 5.6 Problems
- 5.7 Historical remarks

- $6$. Embedding Theorems

- 6.1 The First Embedding Theorem
- 6.2 Refinements of the first embedding theorem
- 6.3 Problems
- 6.4 Historical remarks

- $7$. Models with finite supports

- 7.1 Independence of the Axiom of Choice from the Prime Ideal Theorem
- 7.2 Independence of the Axiom of Choice from the Ordering Principle
- 7.3 Indepencende of the Ordering Principle from the Axiom of Choice for Finite Sets
- 7.4 The Axiom of Choice for Finite Sets
- 7.5 Problems
- 7.6 Historical remarks

- $8$. Some weaker versions of the Axiom of Choice

- 8.1 The Principle of Dependent Choices and its generalization
- 8.2 Independence results concerning the Principle of Dependent Choices
- 8.3 Problems
- 8.4 Historical remarks

- $9$. Nontransferable statements

- 9.1 Statements which imply AC in ZF but are weaker than AC in ZFA
- 9.2 Independence results in ZFA
- 9.3 Problems
- 9.4 Historical remarks

- $10$. Mathematics without choice

- 10.1 Properties of the real line
- 10.2 Algebra without choice
- 10.3 Problems
- 10.4 Historical remarks

- $11$. Cardinal numbers in set theory without choice

- 11.1 Ordering of cardinal numbers
- 11.2 Definability of cardinal numbers
- 11.3 Arithmetic of cardinal numbers
- 11.4 Problems
- 12.5 Historical remarks

- $12$. Some properties contradicting the Axiom of Choice

- 12.1 Measurability of $\aleph_1$
- 12.2 Closed unbounded sets and partition properties
- 12.3 The Axiom of Determinateness
- 12.4 Problems
- 12.5 Historical remarks

- Appendix
- $\text A.1$. Equivalences of the Axiom of Choice
- $\text A.2$. Equivalences of the Prime Ideal Theorem
- $\text A.3$. Various independence results
- $\text A.4$. Miscellaneous examples

- References

- Author index

- Subject index

- List of symbols

## Source work progress

- 1973: Thomas J. Jech:
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