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 (19041908). - 2. 1 König'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 Poincaré and Fréchet. - 2. 5 The German Cantorians. - 2. 6 Father and Son: Julius and Denes König. - 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 (19081918). - 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 (19181940). - 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 ShiftingAxiomatizations for Set Theory. - 4. 10 Consistency and Independence of the Axiom. - 4. 11 Scepticism and Inquiry. - 4. 12 Retrospect and Prospect. - Epilogue: After Gödel. - 5. 1 A Period of Stability: 19401963. - 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. Language: English