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.
Prologue.- 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 Cantor's 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.- 2 Zermelo and His Critics (1904-1908).- 2.1 Konig's "Refutation" of the Continuum Hypothesis.- 2.2 Zermelo's 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. Zermelo's Axiom.- 2.9 Brouwer: A Voice in the Wilderness.- 2.10 Enthusiasm and Mistrust in America.- 2.11 Retrospect and Prospect.- 3 Zermelo's Axiom and Axiomatization in Transition (1908-1918).- 3.1 Zermelo's Reply to His Critics.- 3.2 Zermelo's 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 Hausdorff's Paradox.- 3.8 An Abortive Attempt to Prove the Axiom of Choice.- 3.9 Retrospect and Prospect.- 4 The Warsaw School, Widening Applications, Models of Set Theory (1918-1940).- 4.1 A Survey by Sierpi?ski.- 4.2 Finite, Infinite, and Mediate.- 4.3 Cardinal Equivalents.- 4.4 Zorn's 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 Axiom's 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.- Epilogue: After Godel.- 5.1 A Period of Stability: 1940-1963.- 5.2 Cohen's Legacy.- Conclusion.- Appendix 1 Five Letters on Set Theory.- Appendix 2 Deductive Relations Concerning the Axiom of Choice.- Journal Abbreviations Used in the Bibliography.- Index of Numbered Propositions.- General Index.
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