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Categorical Semantics of Constructive Set Theory

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Categorical Semantics of Constructive Set Theory Categorical semantics of constructive set theory Beim Fachbereich Mathematik der Technischen Universit¨atDarmstadt eingereichte Habilitationsschrift von Benno van den Berg, PhD aus Emmen, die Niederlande 2 Contents 1 Introduction to the thesis 7 1.1 Logic and metamathematics . 7 1.2 Historical intermezzo . 8 1.3 Constructivity . 9 1.4 Constructive set theory . 11 1.5 Algebraic set theory . 15 1.6 Contents . 17 1.7 Warning concerning terminology . 18 1.8 Acknowledgements . 19 2 A unified approach to algebraic set theory 21 2.1 Introduction . 21 2.2 Constructive set theories . 24 2.3 Categories with small maps . 25 2.3.1 Axioms . 25 2.3.2 Consequences . 29 2.3.3 Strengthenings . 31 2.3.4 Relation to other settings . 32 2.4 Models of set theory . 33 2.5 Examples . 35 2.6 Predicative sheaf theory . 36 2.7 Predicative realizability . 37 3 Exact completion 41 3.1 Introduction . 41 3 4 CONTENTS 3.2 Categories with small maps . 45 3.2.1 Classes of small maps . 46 3.2.2 Classes of display maps . 51 3.3 Axioms for classes of small maps . 55 3.3.1 Representability . 55 3.3.2 Separation . 55 3.3.3 Power types . 55 3.3.4 Function types . 57 3.3.5 Inductive types . 58 3.3.6 Infinity . 60 3.3.7 Fullness . 61 3.4 Exactness and its applications . 63 3.5 Exact completion . 66 3.6 Stability properties of axioms for small maps . 73 3.6.1 Representability . 74 3.6.2 Separation . 74 3.6.3 Power types . 74 3.6.4 Function types . 78 3.6.5 Inductive types . 80 3.6.6 Infinity . 87 3.6.7 Fullness . 87 3.7 Soundness . 91 3.8 Completeness . 96 3.9 Set-theoretic axioms . 100 3.10 Positive Heyting categories . 101 4 Realizability 103 4.1 Introduction . 103 4.2 The category of assemblies . 109 4.3 The predicative realizability category . 117 4.4 Additional axioms . 120 4.5 Realizability models for set theory . 127 CONTENTS 5 4.6 A model of CZF in which all sets are subcountable . 131 4.7 Set-theoretic axioms . 137 4.7.1 Axioms of IZF . 137 4.7.2 Axioms of CZF . 138 4.7.3 Constructivist principles . 138 4.8 Predicative categories with small maps . 139 5 Sheaves 143 5.1 Introduction . 143 5.2 Preliminaries . 147 5.2.1 Review of Algebraic Set Theory . 147 5.2.2 W-types . 152 5.2.3 Fullness . 155 5.2.4 Axiom of multiple choice . 157 5.2.5 Main results . 158 5.3 Presheaves . 159 5.3.1 Pointwise small maps in presheaves . 159 5.3.2 Locally small maps in presheaves . 163 5.4 Sheaves . 172 5.4.1 Sheafification . 173 5.4.2 Small maps in sheaves . 175 5.4.3 Collection and fullness in sheaves . 181 5.4.4 W-types in sheaves . 186 5.5 Sheaf models of constructive set theory . 192 6 Applications: derived rules 195 6.1 Introduction . 195 6.2 Constructive set theory . 197 6.2.1 CZF . 197 6.2.2 Inductive definitions in CZF . 198 6.3 Predicative locale theory . 199 6.3.1 Formal spaces . 199 6 CONTENTS 6.3.2 Inductively generated formal topologies . 200 6.3.3 Formal Cantor space . 201 6.3.4 Formal Baire space . 202 6.3.5 Points of a formal space . 204 6.3.6 Morphisms of formal spaces . 206 6.3.7 Double construction . 207 6.4 Sheaf models . 208 6.4.1 Choice principles . 212 6.4.2 Brouwer ordinals . 213 6.5 Main results . 217 7 Note on the Axiom of Multiple Choice 221 7.1 Introduction . 221 7.2 The Set Compactness Theorem . 223 7.3 Stability under exact completion . 227 7.4 Stability under realizability . 230 7.5 Stability under presheaves . 232 7.6 Stability under sheaves . 234 7.7 Relation of AMC to other axioms . 238 7.8 Conclusion . 240 8 Ideas on constructive set theory 241 8.1 Constructive set theory: the very idea . 241 8.2 Constructive set theory as a constructive formal theory . 244 8.3 Constructive set theory as a foundation for formal topology . 245 8.4 Constructive set theory as a predicative formal theory . 247 8.5 The proof theory of constructive set theory . 249 Chapter 1 Introduction to the thesis 1.1 Logic and metamathematics Many different topics are studied within mathematical logic, but for the purposes of this thesis we may say that logic is concerned with mathematical proofs. Although every mathematician knows what a mathematical proof is, it might be good to spend a few more words on the topic. When a mathematician publishes a proof, he1 typically takes a few known the- orems and uses generally accepted proof principles to derive from them a new, un- known, result. The theorems on which he bases his argument have been obtained in the same way and if one would start to trace back what the assumptions are on which these in turn are based, then ultimately one will find a few fundamental axioms con- cerning sets on which all mathematical results are based. It has been laid down very precisely which fundamental set-theoretic axioms are allowed to occur at the end, and which proof principles may be used to derive results from them. So precisely in fact, that they can be captured in a formal system and be studied mathematically: this is what is done in “metamathematics.” The formal system which captures the commonly accepted axioms and methods of proof has been called ZFC, for Zermelo-Fraenkel set theory with the Axiom of Choice. Since this system is an (almost) universally acknowledged foundation for mathematics, one obtains insights into the strengths and limitations of the usual methods of proof by studying ZFC. Two classic results in this respect are: • G¨odel’sIncompleteness Theorems (1931). G¨odel’sFirst Incompleteness Theo- rem implies that ZFC is incomplete, meaning that there are statements that ZFC can neither prove nor refute (unless it is inconsistent, i.e., proves con- tradictory statements). G¨odel’sSecond Incompleteness Theorem shows that a formal statement expressing the consistency of ZFC in ZFC is an example of a statement which ZFC cannot decide. 1Throughout this introduction, please read “she or he” for “he” and “her or his” for “his”. 7 8 CHAPTER 1. INTRODUCTION TO THE THESIS • The independence of the continuum hypothesis. The work of G¨odel(published in 1940) on the constructible hierarchy and that of Cohen (published in 1963) on forcing together imply that the continuum hypothesis is another example of a statement which cannot be decided in ZFC. (The continuum hypothesis says that every subset of the real line is either countable or has the same cardinality as the real line.) 1.2 Historical intermezzo Nowadays, the idea that all of mathematics can be captured in a formal system and that there are a few basic set-theoretic assumptions from which it all follows using only logic, may seem obvious. Nevertheless, it was not always like this and this state of affairs was the result of a long and complicated development which took place during the second half of the nineteenth and the first half of.
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