Conceptions of Set and the Foundations of Mathematics

Description

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics. Contents List of Figures List of Tables Preface 1 Concepts and Conceptions 1.1 Theories 1.2 The Concept of Set 1.3 Criteria of Application and Criteria of Identity 1.4 Extensionality 1.5 What Conceptions Are 1.6 What Conceptions Do 1.7 What Conceptions Are For 1.8 Logical and Combinatorial Conceptions Appendix 1.A Cardinals and Ordinals Appendix 1.B Cantor’s Theorem 2 The Iterative Conception 2.1 The Cumulative Hierarchy 2.2 Iterative Set Theories 2.3 Priority of Construction 2.4 Metaphysical Dependence 2.5 Structuralism and Dependence 2.6 The Minimalist Account 2.7 Inference to the Best Conception 2.8 Conclusion 3 Challenges to the Iterative Conception 3.1 The Missing Explanation Objection 3.2 The Circularity Objection 3.3 The No Semantics Objection 3.4 Higher-Order Semantics 3.5 Kreisel’s Squeezing Argument 3.6 The Status of Replacement 3.7 Conclusion 4 The Naïve Conception 4.1 Paraconsistency and Dialetheism 4.2 Neither Weak nor Trivial 4.3 The Material Strategy 4.4 The Relevant Strategy 4.5 The Model-Theoretic Strategy 4.6 Conclusion 5 The Limitation of Size Conception 5.1 Consistency Maxims 5.2 Cantor Limitation of Size 5.3 Ordinal Limitation of Size 5.4 Von Neumann Limitation of Size 5.5 Frege-Von Neumann Set Theory 5.6 The Extension of Big Properties Objection 5.7 The Arbitrary Limitation Objection 5.8 The No Complete Explanation of the Paradoxes Objection 5.9 The Definite Conception 5.10 Conclusion Appendix 5.A Generalizing McGee’s Theorem 6 The Stratified Conception 6.1 The Early History of Syntactic Restrictionism 6.2 New Foundations and Cognate Systems 6.3 Rejecting Indefinite Extensibility 6.4 The Received View on NF 6.5 From Type Theory to NF 6.6 The Stratified Conception 6.7 NF As a Theory of Logical Collections 6.8 The No Intuitive Model Objection 6.9 The Conflict with Mathematical Practice Objection 6.10 Conclusion 7 The Graph Conception 7.1 Depicting Sets with Graphs 7.2 Four Non-Well-Founded Set Theories 7.3 The Graph Conception of Set 7.4 The Graph Conception and AFA 7.5 The Graph Conception and ZFA 7.6 The No New Isomorphism Types Objection 7.7 The No Intuitive Model Objection 7.8 The No Place for Urelemente Objection 7.9 The No Autonomy Objection 7.10 Conclusion 8 Concluding Remarks Bibliography Index