Comprehensive in its selection of topics and results, this self-contained text examines the relative strengths and consequences of the axiom of choice. Each chapter contains several problems, graded according to difficulty, and concludes with some historical remarks.
An introduction to the use of the axiom of choice is followed by explorations of consistency, permutation models, and independence. Subsequent chapters examine embedding theorems, models with finite supports, weaker versions of the axiom, and nontransferable statements. The final sections consider mathematics without choice, cardinal numbers in set theory without choice, and properties that contradict the axiom of choice, including the axiom of determinacy and related topics
Author(s): Thomas J. Jech
Series: Dover Books on Mathematics
Publisher: Dover Publications
Year: 2008
Language: English
Pages: 213
Cover
Title Page
Copyright Page
Dedication
Preface
Table of Contents
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 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 Prime Ideal Theorem from the Ordering Principle
7.3. Independence 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
11.5. Historical remarks
12. Some properties contradicting the Axiom of Choice
12.1. Measurability of N1
12.2. Closed unbounded sets and partition properties
12.3. The Axiom of Determinateness
12.4. Problems
12.5. Historical remarks
Appendix
A. 1. Equivalents of the Axiom of Choice
A.2. Equivalents of the Prime Ideal Theorem
A.3. Various independence results
A.4. Miscellaneous examples
References
Author index
Subject index
