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Zermelo's Axiom of Choice : Its Origins, Development, and Influence

Author: Gregory H Moore
Publisher: New York, NY : Springer New York, 1982.
Series: Studies in the history of mathematics and physical sciences, 8.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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.  Read more...
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Details

Genre/Form: Electronic books
Additional Physical Format: Printed edition
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Gregory H Moore
ISBN: 9781461394808 1461394805 9781461394785 1461394783
OCLC Number: 852789297
Language Note: English.
Description: 1 online resource (volumes)
Contents: 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 --
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 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.
Series Title: Studies in the history of mathematics and physical sciences, 8.
Responsibility: by Gregory H. Moore.

Abstract:

This book grew out of my interest in what is common to three disciplines: mathematics, philosophy, and history. Though Zermelo's research has provided the focus for this book, much of it is devoted  Read more...

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