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.
Author(s): Luca Incurvati
Publisher: Cambridge University Press
Year: 2020
Language: English
Pages: 255
Contents......Page 6
List of Figures......Page 9
List of Tables......Page 10
Preface......Page 12
1.1 Theories......Page 18
1.2 The Concept of Set......Page 19
1.3 Criteria of Application and Criteria of Identity......Page 24
1.4 Extensionality......Page 27
1.5 What Conceptions Are......Page 29
1.6 What Conceptions Do......Page 31
1.7 What Conceptions Are For......Page 38
1.8 Logical and Combinatorial Conceptions......Page 48
Appendix 1.A Cardinals and Ordinals......Page 49
Appendix 1.B Cantor’s Theorem......Page 51
2 The Iterative Conception......Page 53
2.1 The Cumulative Hierarchy......Page 54
2.2 Iterative Set Theories......Page 61
2.3 Priority of Construction......Page 68
2.4 Metaphysical Dependence......Page 70
2.5 Structuralism and Dependence......Page 76
2.6 The Minimalist Account......Page 78
2.7 Inference to the Best Conception......Page 81
2.8 Conclusion......Page 86
3 Challenges to the Iterative Conception......Page 87
3.1 The Missing Explanation Objection......Page 88
3.2 The Circularity Objection......Page 95
3.3 The No Semantics Objection......Page 98
3.4 Higher-Order Semantics......Page 101
3.5 Kreisel’s Squeezing Argument......Page 104
3.6 The Status of Replacement......Page 107
3.7 Conclusion......Page 117
4.1 Paraconsistency and Dialetheism......Page 118
4.2 Neither Weak nor Trivial......Page 120
4.3 The Material Strategy......Page 122
4.4 The Relevant Strategy......Page 128
4.5 The Model-Theoretic Strategy......Page 138
4.6 Conclusion......Page 143
5.1 Consistency Maxims......Page 145
5.2 Cantor Limitation of Size......Page 151
5.3 Ordinal Limitation of Size......Page 155
5.4 Von Neumann Limitation of Size......Page 156
5.5 Frege-Von Neumann Set Theory......Page 158
5.6 The Extension of Big Properties Objection......Page 162
5.7 The Arbitrary Limitation Objection......Page 163
5.8 The No Complete Explanation of the Paradoxes Objection......Page 165
5.9 The Definite Conception......Page 169
5.10 Conclusion......Page 173
Appendix 5.A Generalizing McGee’s Theorem......Page 174
6.1 The Early History of Syntactic Restrictionism......Page 177
6.2 New Foundations and Cognate Systems......Page 179
6.3 Rejecting Indefinite Extensibility......Page 184
6.4 The Received View on NF......Page 186
6.5 From Type Theory to NF......Page 187
6.6 The Stratified Conception......Page 190
6.7 NF As a Theory of Logical Collections......Page 194
6.8 The No Intuitive Model Objection......Page 196
6.9 The Conflict with Mathematical Practice Objection......Page 198
6.10 Conclusion......Page 199
7 The Graph Conception......Page 201
7.1 Depicting Sets with Graphs......Page 203
7.2 Four Non-Well-Founded Set Theories......Page 204
7.3 The Graph Conception of Set......Page 209
7.4 The Graph Conception and AFA......Page 212
7.5 The Graph Conception and ZFA......Page 218
7.6 The No New Isomorphism Types Objection......Page 226
7.7 The No Intuitive Model Objection......Page 227
7.8 The No Place for Urelemente Objection......Page 230
7.9 The No Autonomy Objection......Page 231
7.10 Conclusion......Page 233
8 Concluding Remarks......Page 235
Bibliography......Page 238
Index......Page 250
