This book grew out of my interest in what is common to three disciplines: mathematics, philosophy, and history. The origins of Zermelo's Axiom of Choice, as well as the controversy that it engendered, certainly lie in that intersection. Since the time of Aristotle, mathematics has been concerned alternately with its assumptions and with the objects, such as number and space, about which those assumptions were made. In the historical context of Zermelo's Axiom, I have explored both the vagaries and the fertility of this alternating concern. Though Zermelo's research has provided the focus for this book, much of it is devoted to the problems from which his work originated and to the later developments which, directly or indirectly, he inspired. A few remarks about format are in order. In this book a publication is indicated by a date after a name; so Hilbert 1926, 178 refers to page 178 of an article written by Hilbert, published in 1926, and listed in the bibliography.
Author(s): Gregory H. Moore
Series: Studies in the History of Mathematics and Physical Sciences 8
Edition: 1
Publisher: Springer
Year: 1982
Language: English
Pages: 412
Cover
Zrnst Zermelo Picture
Title Page
Copyright Page
Dedication Page
Preface
Acknowledgemens
Table of Contents
List of Symbols
Prologue
Chapter 1 The Prehistory of the Axiom of Choice
1.1 Introduction
1.2 The Origins of the Assumption
1.3 The Boundary between the Finite and the Infinite
1.4 Cantors Legacy of Implicit Uses
1.5 The Well-Ordering Problem and the Continuum Hypothesis
1.6 The Reception of the Well-Ordering Problem
1.7 Implicit Uses by Future Critics
1.8 Italian Objections to Arbitrary Choices
1.9 Retrospect and Prospect
Chapter 2 Zermelo and His Critics (1904-1908)
2.1 Konigs Refutation of the Continuum Hypothesis
2.2 Zermelos Proof of the Well-Ordering Theorem
2.3 French Constructivist Reaction
2.4 A Matter of Definitions: Richard, Poincare, and Frechet
2.5 The German Cantorians
2.6 Father and Son: Julius and Denes Konig
2.7 An English Debate
2.8 Peano: Logic vs. Zermelos Axiom
2.9 Brouwer: A Voice in the Wilderness
2.10 Enthusiasm and Mistrust in America
2.11 Retrospect and Prospect
Chapter 3 Zermelos Axiom and Axiomatization in Transition (1908-1918)
3.1 Zermelos Reply to His Critics
3.2 Zermelos Axiomatization of Set Theory
3.3 The Ambivalent Response to the Axiomatization
3.4 The Trichotomy of Cardinals and Other Equivalents
3.5 Steinitz and Algebraic Applications
3.6 A Smoldering Controversy
3.7 Hausdorffs Paradox
3.8 An Abortive Attempt to Prove the Axiom of Choice
3.9 Retrospect and Prospect
Chapter 4 The Warsaw School, Widening Applications, Models of Set Theory (1918-1940)
4.1 A Survey by Sierpihski
4.2 Finite, Infinite, and Mediate
4.3 Cardinal Equivalents
4.4 Zorns Lemma and Related Principles
4.5 Widening Applications in Algebra
4.6 Convergence and Compactness in General Topology
4.7 Negations and Alternatives
4.8 The Axioms Contribution to Logic
4.9 Shifting Axiomatizations for Set Theory
4.10 Consistency and Independence of the Axiom
4.11 Scepticism and Inquiry
4.12 Retrospect and Prospect
Chapter 5 Epilogue: After Godel
5.1 A Period of Stability: 1940-1963
5.2 Cohens Legacy
Conclusion
Appendix 1 Five Letters on Set Theory
I. Letter from Hadamard to Borel
II. Letter from Baire to Hadamard
III. Letter from Lebesgue to Borel
IV. Letter from Hadamard to Borel
V. Letter from Borel to Hadamard
Appendix 2 Deductive Relations Concerning the Axiom of Choice
Journal Abbreviations Used in the Bibliography
Bibliography
Index of Numbered Propositions
