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The axiom of choice /
Provability, Computability and Reflection.
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Amsterdam : New York :
North-Holland Pub. Co. ; American Elsevier Pub. Co.,
1973.
|
| Colección: | Studies in logic and the foundations of mathematics ;
v. 75. |
| Temas: | |
| Acceso en línea: | Texto completo Texto completo Texto completo |
Tabla de Contenidos:
- Front Cover; The Axiom of Choice; Copyright Page; Preface; Contents; Preface; Chapter 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; Chapter 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; Chapter 3. Consistency of the Axiom of Choice
- 3.1. Axiomatic systems and consistency3.2. Axiomatic set theory; 3.3. Transitive models of ZF; 3.4. The constructible universe; 3.5. Problems; 3.6. Historical remarks; Chapter 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; Chapter 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 Principle5.6. Problems; 5.7. Historical remarks; Chapter 6. Embedding Theorems; 6.1. The First Embedding Theorem; 6.2. Refinements of the First Embedding Theorem; 6.3. Problems; 6.4. Historical remarks; Chapter 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 remarksChapter 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; Chapter 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; Chapter 10. Mathematics without choice; 10.1. Properties of the real line; 10.2. Algebra without choice; 10.3. Problems; 10.4. Historical remarks
- Chapter 11. Cardinal numbers in set theory without choice11.1. Ordering of cardinal numbers; 11.2. Definability of cardinal numbers; 11.3. Arithmetic of cardinal numbers; 11.4. Problems; 11.5. Historical remarks; Chapter 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