Author | Moore, Gregory H. author |
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Title | Zermelo's Axiom of Choice [electronic resource] : Its Origins, Development, and Influence / by Gregory H. Moore |
Imprint | New York, NY : Springer New York, 1982 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-9478-5 |
Descript | XIV, 412 p. online resource |
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 (1904โ1908) -- 2.1 Kรถnig's โRefutationโ 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, Poincarรฉ, and Frรฉchet -- 2.5 The German Cantorians -- 2.6 Father and Son: Julius and Denes Kรถnig -- 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 (1908โ1918) -- 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 (1918โ1940) -- 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 Gรถdel -- 5.1 A Period of Stability: 1940โ1963 -- 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