by Gregory H. Moore

Prologue

Chapter 1: The Prehistory of the Axiom of Choice

- Introduction
- The Origins of the Assumption
- The Boundary between the Finite and the Infinite
- Cantor's Legacy of Implicit Uses
- The Well-Ordering Problem and the Continuum Hypothesis
- The Reception of the Well-Ordering Problem
- Implicit Uses by Future Critics
- Italian Objections to Arbitrary Choices
- Retrospect and Prospect

Chapter 2: Zermelo and His Critics (1904-1908)

- Koenig's "Refutation" of the Continuum Hypothesis
- Zermelo's Proof of the Well-Ordering Theorem
- French Constructivist Reaction
- A Matter of Definitions: Richard, Poincare, and Frechet
- The German Cantorians
- Father and Son: Julius and Denes Koenig
- An English Debate
- Peano: Logic vs. Zermelo's Axiom
- Brouwer: A Voice in the Wilderness
- Enthusiasm and Mistrust in America
- Retrospect and Prospect

Chapter 3: Zermelo's Axiom and Axiomatization in Transition (1908-1918)

- Zermelo's Reply to His Critics
- Zermelo's Axiomatization of Set Theory
- The Ambivalent Response to the Axiomatization
- The Trichotomy of Cardinals and Other Equivalents
- Steinitz and Algebraic Applications
- A Smoldering Controversy
- Hausdorff's Paradox
- An Abortive Attempt to Prove the Axiom of Choice
- Retrospect and Prospect

Chapter 4: The Warsaw School, Widening Applications, Models of Set Theory (1918-1940)

- A Survey by Sierpinski
- Finite, Infinite, and Mediate
- Cardinal Equivalents
- Zorn's Lemma and Related Principles
- Widening Applications in Algebra
- Convergence and Compactness in General Topology
- Negations and Alternatives
- The Axiom's Contribution to Logic
- Shifting Axiomatizations for Set Theory
- Consistency and Independence of the Axiom
- Scepticism and Inquiry
- Retrospect and Prospect

Epilogue: After Goedel

- A Period of Stability: 1940-1963
- Cohen's Legacy

Conclusion

Appendix I: Five Letters on Set Theory

Appendix II: Deductive Relations Concerning the Axiom of Choice

**From the Preface:**

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.

- The Axiom of Choice and Its Equivalents

This ProvenMath page states the Axiom of Choice in the formal setting of axiomatic set theory together with a number of its equivalents like the Well-Ordering Principle, Hausdorff's Maximal Principle, and Zorn's Lemma. Strict proofs are given for the equivalence of the Axiom of Choice with any of the above-mentioned principles. - Heine continuity implies Cauchy continuity without the Axiom of Choice

On this page we state and prove that every Heine continuous real function is also Cauchy continuous. In our proof we do not use the Axiom of Choice.