Infinity and truth [Workshop on Infinity and Truth was held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011

This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters are by leading experts in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally unsolvable within the Zermelo Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progresses in foundational studies. The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of benefit to students, researchers and mathematicians interested in the foundations of mathematics. Readership: Mathematicians, philosophers, scientists, graduate students, academic institutions, and research organizations interested in logic and the philosophy of mathematics. CONTENTS 6 Foreword 8 Preface 10 Section I. Invited Lectures 6 Absoluteness, Truth, and Quotients 12 1. Finitism, ‘Countablism’ and a Little Bit Further 12 2. Independence 14 2.1. The story of projective sets 15 3. Absoluteness 17 3.1. Beyond projective sets 18 3.2. Absoluteness and the uncountable 20 3.3. Level by level 20 4. Third-Order Arithmetic 21 4.1. Conditional absoluteness 21 4.2. Π21? 23 5. Quotient Borel Structures 25 5.1. Trivial automorphisms 25 5.2. C*-algebras and their multipliers 26 5.3. General rigidity conjectures 27 Acknowledgments 29 Appendix 29 A.1. Hereditary sets 30 A.2. Arithmetical formulas 30 A.3. Analytical formulas 30 A.4. Examples 31 References 33 A Multiverse Perspective on the Axiom of Constructibility 36 1. Introduction 36 2. Some New Problems with Maddy’s Proposal 39 3. Several Ways in which V = L is Compatible with Strength 44 4. An Upwardly Extensible Concept of Set 51 References 55 Hilbert, Bourbaki and the Scorning of Logic 58 1917: Hilbert returns to the foundations of mathematics 62 Some terminology 63 1928: publication of the treatise of Hilbert and Ackermann 65 Logic in the twenties 65 1922: the Hilbert operator is launched 66 1928: Hilbert at Bologna 68 1928: the war of the Frogs and the Mice 68 1929: the completeness theorem 69 1930/31: the incompleteness theorems 69 1931/34: Hilbert’s delayed response to the incompleteness theorems 70 Hilbert’s programme after Godel 71 1934, 1939: publication in two volumes of the treatise of Hilbert and Bernays 72 1935: the naissance of Bourbaki 73 Bourbaki’s syntax 75 The length of τ-expansions 76 Every null term is equal to a proper term 77 Any two null terms are equal 77 Perverted interpretation of quantifiers 79 Discussion 79 Bourbaki’s remarks on progress in logic 81 Bourbaki’s account of the incompleteness theorem 82 Godement’s formal system 83 Godement’s set-theoretic axioms 86 Misunderstandings of work of logicians 94 Unease in the presence of logic 95 page 7: the problem of choice 117 Discussion of equality 121 A list of axioms of set theory 121 The legacy of Napoleon: the foundation of the modern French university system 132 Politics and mathematics 134 Bourbaki and French nationalism 139 The chimera of completeness 139 La Tribu 141 Bourbaki consult Rosser 147 Why use Bourbaki’s formalisation? 150 Structuralism: a part but not the whole of mathematics 151 Another collapse 154 Back to St Benedict 156 Acknowledgments 159 References 160 Toward Objectivity in Mathematics 168 1. Objectivity and Objectivism 168 2. Mathematics as Part of Human Knowledge 170 3. Set Theory and the Unity of Mathematics 172 4. Set-Theoretic Realism 174 4.1. An epistemological question 174 4.2. The intrinsicist answer 175 4.3. The “testable consequences” answer 175 4.4. The Thin Realist answer 176 5. Insights from Reverse Mathematics 177 6. Wider Cultural Significance? 179 Acknowledgment 179 References 180 Sort Logic and Foundations of Mathematics 182 1. Introduction 182 2. Sort Logic 185 2.1. Basic concepts 185 2.2. Syntax 185 2.3. Axioms 187 2.4. Semantics 188 3. Sort Logic and Set Theory 191 4. Sort Logic and Foundations of Mathematics 192 References 196 Reasoning about Constructive Concepts 198 1. 198 2. 200 3. 201 4. 203 5. 205 6. 206 References 209 Perfect Infinities and Finite Approximation 210 1. Introduction 210 2. Continuity and its Alternatives 211 2.1. 211 2.2. 211 2.3. 211 2.4. 212 3. In Search of Logically Perfect Structures 213 3.1. 213 3.2. 215 3.3. 215 3.4. Topological structures 217 3.5. 218 3.6. 218 3.7. 219 3.8. 220 4. Structural Approximation 222 4.1. Topological structures 222 4.2. Structural approximation 223 5. Examples 225 5.1. Metric spaces 225 5.2. Cyclic groups in profinite topology 226 5.3. The ring of p-adic integers 226 5.4. Compactified groups 227 5.5. Cyclic groups in metric topology and their compactifications 228 5.6. 2-ends compactification of Z 228 6. Approximation by Some Finite Structures 230 6.1. Approximation by finite fields 230 6.2. Approximation by finite groups 232 References 233 Section II. Special Session 6 An Objective Justification for Actual Infinity? 236 1. Introduction 236 2. Objectivity in Mathematics 237 3. Potential Infinity versus Actual Infinity 237 4. Insights from Reverse Mathematics 238 References 239 Oracle Questions 240 1. Introduction 240 2. Questions 241 2.1. Ilijas Farah 241 2.2. Moti Gitik 241 2.3. Joel David Hamkins 241 2.4. Juliette Kennedy 242 2.5. Steffen Lempp 243 2.6. Stephen G. Simpson 243 2.7. Theodore Slaman 243 2.8. Jouko Vaananen 243 2.9. Nik Weaver 244 2.10. W. Hugh Woodin 244 2.11. Boris Zilber 245 References 245 This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters cover topics in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally unsolvable within the Zermelo-Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progress in foundational studies.The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of interest to students, researchers and mathematicians concerned with issues in the foundations of mathematics. Section I. Invited Lectures. Absoluteness, Truth, And Quotients / Ilijas Farah -- A Multiverse Perspective On The Axiom Of Constructibility / Joel David Hamkins -- Hilbert, Bourbaki And The Scorning Of Logic / A.r.d. Mathias -- Toward Objectivity In Mathematics / Stephen G. Simpson -- Sort Logic And Foundations Of Mathematics / Jouko Väänänen -- Reasoning About Constructive Concepts / Nik Weaver -- Perfect Infinities And Finite Approximation / Boris Zilber -- Section Ii. Special Session. An Objective Justification For Actual Infinity? / Stephen G. Simpson -- Oracle Questions / Theodore Slaman And W. Hugh Woodin. Chitat Chong, Qi Feng, Theodore A. Slaman, W. Hugh Woodin. Based On The Talks Given At The Workshop On Infinity And Truth Held At The Institute For Mathematical Sciences, National University Of Singapore, From 25 To 29 July 2011. Includes Bibliographical References.