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Florida State University Libraries Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2009 A Defense of Platonic Realism in Mathematics: Problems About the Axiom of Choice Wataru Asanuma Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES A DEFENSE OF PLATONIC REALISM IN MATHEMATICS: PROBLEMS ABOUT THE AXIOM OF CHOICE By WATARU ASANUMA A Dissertation submitted to the Department of Philosophy in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Spring Semester, 2009 The members of the Committee approve the Dissertation of Wataru Asanuma defended on April 7, 2009. __________________________ Russell M. Dancy Professor Directing Dissertation __________________________ Philip L. Bowers Outside Committee Member __________________________ J. Piers Rawling Committee Member __________________________ Joshua Gert Committee Member Approved: _________________________________________________________________ J. Piers Rawling, Chair, Philosophy _________________________________________________________________ Joseph Travis, Dean, College of Arts and Sciences The Graduate School has verified and approved the above named committee members. ii ACKNOWLEDGMENTS First of all, I would like to thank all my dissertation committee members, Dr. Russell Dancy, Dr. Piers Rawling, Dr. Joshua Gert and Dr. Philip Bowers. Special thanks go to my major professor, Dr. Dancy, who has guided my dissertation every step of the way. The courses they offered, especially Dr. Dancy‘s ―υlato‘s ‗Unwritten Doctrines‘,‖ Dr. Rawling‘s ―Modern Logic I & II‖ and Dr. Gert‘s ―υhilosophy of Mathematics,‖ formed the backbone of my dissertation. I would also like to thank Dr. Bowers (Department of Mathematics) for serving as an outside committee member. It was exceptionally fortunate that I had an opportunity to present the parts of my dissertation at the 2008 Logic, Mathematics and Physics Graduate Philosophy Conference at the University of Western Ontario. The conference has marked a major turning point for me, and continues to inspire and motivate further research. Especially I‘m very grateful to Dr. John Bell and Dr. William Harper for their kind and insightful words on my presentation. In addition, I had opportunities to present the parts of my dissertation at the 7th Hawaii International Conference on Arts and Humanities and at the 12th Northeast Florida Student Philosophy Conference. I would like to express my appreciation to the great audience for their attention. iii TABLE OF CONTENTS List of Figures .................................................................................................................... vi Abstract ............................................................................................................................ viii INTRODUCTION ........................................................................................................... 1 1. WHAT IS THE AXIOM OF CHOICE? ...................................................................... 4 1.1 The Axioms of ZF Set Theory .......................................................................... 4 1.2 The Axiom of Choice and its Equivalents ........................................................ 5 1.3 The Consequences of the Axiom of Choice ...................................................... 8 1.4 A Weaker Form of the Axiom of Choice .......................................................... 11 1.5 The Axiom of Choice and the Continuum Hypothesis ..................................... 13 1.6 Fictionalism or Instrumentalism ....................................................................... 15 2. LEBESGUE‘S THEORY OF MEASURE .................................................................... 20 2.1 Lebesgue‘s Theory of Integration ..................................................................... 20 2.2 Measurable Cardinals ........................................................................................ 31 3. THE BANACH-TARSKI PARADOX ........................................................................ 39 3.1 Preliminaries ..................................................................................................... 40 3.2 Non-Lebesgue Measurable Sets ........................................................................ 45 3.3 The Hausdorff Paradox ................................................................................... 47 3.4 The Banach-Tarski Paradox .............................................................................. 55 3.5 What is a Paradox? ............................................................................................ 58 3.6 What cannot Happen? ....................................................................................... 63 3.7 A Paradox without the Axiom of Choice? ......................................................... 64 3.8 Some Philosophical Implications ...................................................................... 66 4. GÖDEL‘S INCOMPLETENESS THEOREMS .......................................................... 69 4.1 Gödel‘s First Incompleteness Theorem ............................................................. 69 4.2 Turing Machines ............................................................................................... 73 4.3 Recursive Functions and Recursive Sets .......................................................... 78 4.4 The Halting Problem ......................................................................................... 82 4.5 The Undecidability of First-order Logic and Arithmetic .................................. 85 4.6 Undecidability and Incompleteness .................................................................. 87 4.7 Gödel‘s Second Incompleteness Theorem ........................................................ 90 4.8 Relative Consistency Proofs ............................................................................. 91 iv 5. MODEL-THEORETIC ARGUMENTS ...................................................................... 95 5.1 What is a Model? .............................................................................................. 96 5.2 The Löwenheim-Skolem Theorem and the Skolem Paradox ........................... 97 5.3 Quine‘s Thesis of the Indeterminacy of Translation ......................................... 103 5.4 υutnam‘s Model-Theoretic Arguments against Metaphysical Realism ............ 107 5.5 The Lessons from the Model-Theoretic Arguments ......................................... 111 6. WHAT IS MATHEMATICAL EXISTENCE? .................................................................... 116 6.1 ψenacerraf‘s ωhallenges to Platonism .............................................................. 116 6.2 Some Applications of the Axiom of Choice ..................................................... 119 6.3 The Axiom of Determinacy ............................................................................... 120 6.4 The Axiom of Constructibility .......................................................................... 121 6.5 Anselm‘s Argument of the Existence of God .................................................... 124 6.6 Locke on Essences ............................................................................................ 126 6.7 Essence and Existence in Mathematics ............................................................. 130 6.8 What is a Maximally Consistent Theory? ......................................................... 131 CONCLUSION ............................................................................................................... 136 APPENDIX ....................................................................................................................... 138 BIBLIOGRAPHY ............................................................................................................. 140 BIOGRAPHICAL SKETCH ............................................................................................. 148 v LIST OF FIGURES Figure 1: The Axiom of Choice ......................................................................................... 6 Figure 2: Riemann Integral vs. Lebesgue Integral ............................................................ 28 Figure 3: Example of a partially ordered set ..................................................................... 40 Figure 4: Partial order by inclusion of the power set of a set S＝{0, 1, 2} ....................... 43 Figure 5: The use of the Axiom of Choice in the proof of Zorn‘s Lemma ........................ 44 Figure 6: Example of a non-Lebesgue measurable set ...................................................... 47 Figure 7: G-paradoxical ..................................................................................................... 48 Figure 8: G-equidecomposable .......................................................................................... 48 Figure 9: The Hausdorff Paradox ...................................................................................... 50 Figure 10: The existence of a set T containing exactly one element from each G-orbit ... 52 Figure 11: Hausdorff‘s paradoxical decomposition........................................................... 53 Figure 12: The Weak Form of the Banach-Tarski Paradox (Two Spheres from One Version) ...................................................................................... 57
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