Book:Thomas J. Jech/The Axiom of Choice

Subject Matter

 * Set Theory
 * The Axiom of Choice

Contents

 * Preface (December 1972)


 * $1$.
 * 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


 * $2$.


 * 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


 * $3$.


 * 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


 * $4$.


 * 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


 * $5$.


 * 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


 * $6$.


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


 * $7$.


 * 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


 * $8$.


 * 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


 * $9$.


 * 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


 * $10$.


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


 * $11$.


 * 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


 * $12$.


 * 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


 * $\text A.1$. 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
 * $\text A.4$. Miscellaneous examples











Source work progress
* : $1.3$ A paradoxical decomposition of the sphere: Theorem $1.2$