Book:Thomas J. Jech/The Axiom of Choice

Thomas J. Jech: The Axiom of Choice

Published $\text {1973}$, Dover Publications

ISBN 0-486-46624-8

Subject Matter

• Set Theory
• The Axiom of Choice

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

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