The Axiom of Choice Studies in Logic and the Foundations of Mathematics, Vol. 75

[from the Preface]The book was written in the long Buffalo winter of 1971-72. It is an attempt to show the place of the Axiom of Choice in contemporary mathematics. Most of the material covered in the book deals with independence and relative strength of various weaker versions and consequences of the Axiom of Choice. Also included are some other results that I found relevant to the subject.The selection of the topics and results is fairly comprehensive, nevertheless it is a selection and as such reflects the personal taste of the author. So does the treatment of the subject. The main tool used throughout the text is Cohen’s method of forcing. I tried to use as much similarity between symmetric models of ZF and permutation models of ZFA as possible. The relation between these two methods is described in Chapter 6, where I present the Embedding Theorem (Jech-Sochor) and the Support Theorem (Pincus).Consistency and independence of the Axiom of Choice (due to Godel and Cohen, respectively) are presented in Chapters 3 and 5. Chapter 4 introduces the permutation models (Fraenkel, Mostowski, Specker). Chapters 2 and 10 are devoted to the use of the Axiom of Choice in mathematics. Chapter 2 contains examples of proofs using the Axiom of Choice, whereas Chapter 10 provides counterexamples in the absence of the Axiom of Choice.Chapters 7 and 8 deal with various consequences of the Axiom of Choice.In Chapter 9 we discuss the differences between permutation models of ZFA and symmetric models of ZF.Chapter 11 is devoted to questions concerning cardinal numbers in set theory without the Axiom of Choice. Finally, in Chapter 12 we discuss briefly the Axiom of Determinacy and related topics. This subject has been recently in the focus of attention, and it seems that solution of the open problems presented at the end of Chapter 12 would mean an important contribution to the investigation of the Foundations of Mathematics.The book is more or less self-contained, with proofs of a sort 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 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