The Palgrave Companion to the Philosophy of Set Theory

Description

This volume showcases some of the up-and-coming voices of an emerging field - the philosophy of set theory - which in recent years has gained prominence in the philosophy of mathematics. The chapters in this volume both present new topics and propose solutions to old problems. It contains a broad picture of the philosophy of set theory, examining questions from epistemology and ontology, whilst touching on the use of formal theories in the study of mathematical infinity. Key features of this volume: • Explores new and interesting connections between philosophy, set theory, and the study of infinity. • Considers questions intended to appeal to a wider audience in both philosophy and mathematical logic. • Examines three key areas of study: Epistemology, Formal Theories, and Ontology. The book provides a key reference text for future debates and is ideal for both newcomers to the philosophy of set theory and established researchers in the field. Contents Notes on Contributors List of Table 1 Introduction 1 Overview of the Volume 2 Overview of the Chapters Part I Epistemology 2 The Problem of Existence for Descriptivism About the Reference of Set-Theoretic Expressions 1 Introduction 2 Descriptivism 3 The Problem of Existence 4 Conclusion References 3 Intuition and Observation 1 Informal and Formal Proof 2 Knowing the Axioms 3 Justification and Reliability 4 Science and Mathematics 5 The Problem of Conflicting Intuitions 6 Theory Ladenness 7 Heretics 8 A Misleading Contrast? 9 Mathematics as Philosophy 10 Illustration: Modal Metaphysics 11 Disagreement, Objectivity, and Practice 12 Implications for Physics 13 Armchair Pluralism Bibliography 4 Arealism, Thin Realism, and the Problem of Extrinsic Evidence 1 The Problem of Extrinsic Evidence 2 Three Recent Responses to the Problem 3 An Arealist Solution to the Problem References 5 The Hidden Use of New Axioms 1 Introduction 2 Studying the Roles of Axioms in Mathematical Practice 3 Data on Large Cardinal Axioms in Set-Theoretic Practice 3.1 Sample Set 3.2 Results 4 Discussion: The Hidden Use of Large Cardinal Axioms 4.1 Conceptualisation 4.2 Exemplary Proofs 4.2.1 Borel Determinacy 4.2.2 Cichoń's Maximum 4.3 Significance of Hidden Use for Set-Theoretic Practitioners 4.3.1 Extrinsic Justification of New Axioms 4.3.2 Pragmatic Significance 5 Conclusion References Part II Formal Theories 6 The Iterative Notion of Function and the Iterative Notion of Set 1 Sets as Functions/Functions as Sets 2 The Iterative Notion of Function 3 Functional-Level Theory 3.1 Notational Preliminaries 3.2 Using Stages: FST 3.3 Eliminating Stages: FLT 3.4 FLT is Quasi-Categorical 3.5 Adding More Functions 3.6 Synonymy 4 Function-Theoretic Judicial Foundations 4.1 Maddy on Judicial Foundations 4.2 Consistency and Coherence 4.3 Function-Theoretic Foundations 4.4 Adding New Axioms 4.5 The Same Judicial Foundation 5 Joint-Carving Metaphysical Foundations 5.1 Monistic Joint-Carving and Its Surds 5.2 Pluralistic Joint-Carving and Its Surds 6 Metaphysical Foundations Without Joint-Carving? 7 Conclusion, and Putnam Revisited A Background on LT B Elementary Considerations Regarding FLT B.1 Formalizing Partial Functions B.2 Key Definitions and Axioms B.3 Key Results B.4 FLT and Categories C Autonomous Judicial Foundations in FLT References 7 What Set Theory Could Not Be About 1 Refining the Claim 1.1 What Is Set Theory in Its Capacity as a Foundation for Mathematics? 1.2 What Would It Mean for the Content of Set Theory to Be Specified? 1.3 The Argument 2 Defending the Claim 2.1 An Alternative Interpretation 2.2 Some Alternative Languages 2.2.1 Function Theory 2.2.2 ETCS+ 2.2.3 MV 2.3 Summary 3 Reviewing Some Consequences 3.1 Conclusion Appendix References 8 Hamkins' Analogy Between Set Theory and Geometry: Pluralism by Leveling Up? 1 Introduction 2 Background 2.1 Hamkins' Platonist Multiverse 2.2 Hamkins' Analogy 3 Geometrical Pluralism by Leveling Up 4 Is Hamkins' Multiverse Set-Theoretic Pluralism via Leveling Up? 5 What Might Set-Theoretic Pluralism via Leveling Up Look Like? 5.1 Basic Proposal for Leveling Up Pluralism About Set Theory 5.2 Prior Work on Counterpossible Counterfactuals 5.3 Features of This View 6 Conclusion References 9 Basic Mathematics Beyond Classicality 1 Introduction 2 Non-classical Set Theories and Their Models 2.1 Generalized Algebra-Valued Models 2.2 Negation-Free Fragment 2.3 Reasonable Implication Algebra 3 Mathematics in the Models of Non-classical Set Theories 3.1 The Quine-Putnam Indispensability Argument 4 Independence and the Interpretations of & = 4.1 Independence Results in Non-classical Set Theories 4.2 Interpretations of and = 5 Conclusion References Part III Ontology 10 Iteration and Dependence Again 1 The Cumulative Hierarchy Structure 2 Distinguishing Between Two Debates 3 Structuralism and Dependence 4 Structuralism and Indefinite Extensibility 5 Conclusion References 11 What Is Forcing Potentialism? 1 Potentialism 2 The Iterative Conception of Set 3 The Challenge for Forcing Potentialism 3.1 Convergent vs. Branching Extensions 3.2 The Iterative Conception and Convergence 3.3 Objections 3.3.1 What Matters Is the Logic 3.3.2 The Iterative Conception Is Consistent with Branching Extensions 3.3.3 Countable Models Are Non-standard 3.3.4 Some Forcing Extensions Are Convergent 3.4 Conclusion 4 What Forcing Potentialism Could Be? 4.1 The Iterative Logical Conception of Set 4.2 Lawless Sequences 4.3 A Model of Height and Width Potentialism 4.3.1 Formal Properties of L 4.3.2 Philosophical Remarks 5 Conclusion A Proofs A.1 Proposition 11.1 A.2 Proposition SPIlinkcolor10011.3 References 12 On the Consistency of Height and Width Potentialism 1 Overview 2 Background 3 Height and Width Potentialism Combined 3.1 Core Logical Principles 3.2 Simple HWP 3.3 Strong HWP 4 Roberts 5 Brauer Appendix Necessary Lemmas Theorems References 13 Higher-Order Platonism and Multiversism 1 Hamkins' Multiverse and Platonism 2 Higher-Order Platonism: Main Features 3 Object Theory 4 Higher-Order Platonism Inside Object Theory 5 Three Problems: Articulation, Skepticism, Practice 5.1 Articulation 5.2 Skepticism 5.3 Practice 6 Concluding Remarks References 14 Language, Models, and Reality: Weak Existence and a Threefold Correspondence 1 Introduction 2 An Analogy: The Use of Models in Science and Set Theory 3 The Circularity Problem 4 Existing Accounts of Weak Existence 4.1 Consistency Implies Existence 4.2 Abstraction Principles 4.3 Internal Questions and Easy Arguments 5 Underwriting Models with Weak Existence 6 Objections 7 Conclusions and Open Questions 7.1 Compatibility Between Different Accounts of Weak Existence 7.2 Verisimilitude of a Model 7.3 Which Theories are ``Legitimate''? 7.4 Characterising Debates and Communication 7.5 The Access Problem 7.6 Modelism 7.7 Weak Existence as a Motivation for Further Axioms? 7.8 Summing Up References Index