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Zermelo’s Axiom of Choice Its Origins, Development, and Influence

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 concerne...

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Bibliographic Details
Main Author: Moore, G.H.
Format: eBook
Language:English
Published: New York, NY Springer New York 1982, 1982
Edition:1st ed. 1982
Series:Studies in the History of Mathematics and Physical Sciences
Subjects:
Mathematical Logic
Mathematical Logic And Foundations
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 4.12 Retrospect and Prospect
  • Epilogue: After Gödel
  • 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
  • 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 ShiftingAxiomatizations for Set Theory
  • 4.10 Consistency and Independence of the Axiom
  • 4.11 Scepticism and Inquiry
  • 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 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

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