Homotopy Type-Theoretic Interpretations of Constructive Set Theories Cesare Gallozzi Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds School of Mathematics July 2018 The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledge- ment. c 2018, The University of Leeds and Cesare Gallozzi The right of Cesare Gallozzi to be identified as author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. To all my teachers. Acknowledgements I wish to thank all those who made this thesis possible. I thank all my teachers and in particular my supervisor Nicola Gambino for his tireless generosity and all the help and guidance he offered during the course of the PhD. I thank the University of Leeds and the School of Mathematics for their financial support. I thank all those who contributed to improve this thesis by answering my ques- tions or making comments on the material: my co-supervisor Michael Rathjen, and also Peter Hancock, John Truss, Stan Wainer, Martin Hofmann, Helmut Schwichtenberg, Michael Toppel, Anton Freund, Andrew Swan, Jakob Vidmar, Nicolai Kraus and Fredrik Nordvall Forsberg. I thank my parents and my grandmother for all their care and for encouraging my interests in science and mathematics since my early childhood. I also thank all my friends. iv Abstract This thesis deals primarily with type-theoretic interpretations of constructive set theories using notions and ideas from homotopy type theory. We introduce a family of interpretations · k;h for 2 ≤ k ≤ 1 and 1 ≤ h ≤ 1 of the set theory BCS into the type theory H,J inK which sets and formulas are inter- preted respectively as types of homotopy level k and h. Depending on the values of the parameters k and h we are able to interpret different theories, like Aczel's CZF and Myhill's CST. We relate the family · k;h to the other interpretations of CST into homotopy type theory already studiedJ K in the literature in [Uni13] and [Gyl16a]. We characterise a class of sentences valid in the interpretations · k;1 in terms of the ΠΣ axiom of choice, generalising the characterisation of [RT06]J K for Aczel's interpretation. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The formulas valid in this interpretation are then characterised in terms of the axiom of unique choice. We extend the analysis of Aczel's interpretation provided in [GA06] to the in- terpretations of CST into homotopy type theory, providing a comparative analysis. This is done formulating in the logic-enriched type theory the key principles used in the proofs of the two interpretations. We also investigate the notion of feasible ordinal formalised in the context of a linear type theory equipped with a type of resources. This type theory was originally introduced by Hofmann in [Hof03]. We disprove Hofmann's conjecture on the definable ordinals, by showing that for any given k 2 N the ordinal !k is definable. v Contents Introduction 1 Background 1 Constructive set theories and type theories 2 Set theories 2 Type theories 2 Homotopy type theories 4 Linear type theories 6 The state of the literature 6 Outline of the thesis and main results 8 Chapter 1. Homotopy type-theoretic interpretations 13 1.1. Type-theoretic preliminaries 14 The type theory H 14 The type theory HoTT 19 1.2. The homotopy type-theoretic interpretations 21 1.3. Interpreting constructive set theories 23 Some lemmas on equality and formulas 24 Validity of the set-theoretic axioms 29 1.4. An alternative interpretation of equality 36 Chapter 2. A characterisation of the interpretations · k;1 41 2.1. Set-theoretic preliminariesJ K 42 2.2. Proof-theoretic preliminaries 43 2.3. Relating the interpretations · 1;1 and · k;1 45 J K J K 2.4. ΠΣ-AC is valid in the interpretations · k;1 48 Injectively presented setsJ K 49 Strong bases 50 2.5. The proof-theoretic characterisation 53 2.6. A side question: ΠΣI-AC and hsets 55 Chapter 3. Equivalence between some interpretations 57 3.1. Preliminaries on setoids and the epi-mono factorisation 58 3.2. Equivalence with Gylterud's interpretation 61 vii viii CONTENTS 3.3. Equivalence with the HoTT book interpretation 65 3.4. Equivalence with the interpretations · k;1 69 3.5. Logical aspectsJ K 71 Chapter 4. A characterisation of the · PHP interpretation 77 4.1. Logic-enriched type theoriesJ K 78 4.2. The propositions-as-hprops interpretation 81 4.3. Characterisation of the formulas valid in · PHP 82 J K From the interpretation · PHP to the PaHP principle 83 Characterisation of the principleJ K PaHP 84 4.4. A logic-enriched type theory for · 1;1 87 Other principles for a logic-enrichedJ K type theory 87 0 Validity of the principles under · PHP 90 J K Chapter 5. Analysis of the interpretations via logic-enriched type theories 93 5.1. The combinatorial interpretation 95 5.2. The hybrid interpretation 100 5.3. Relating logic-enriched type theories 104 5.4. Conclusions 108 Chapter 6. Feasible ordinals 111 6.1. Ordinal notations in LFPL 112 Syntax of LFPL + O 113 6.2. Comparing systems 118 6.3. Definable ordinals 121 6.4. Conclusions 126 Chapter 7. Future research 131 Appendix 133 A. Constructive set theories 133 Axioms of BCS 133 Axioms of CST 134 Axioms of CZF 134 The Fullness axiom 134 B. Type-theoretic rules 136 Martin-L¨oftype theory 136 Logic-enriched type theories 141 C. Set-theoretic semantics of LFPL + O 143 Bibliography 147 Introduction Background The topic of this thesis belongs to the general area of the study of the found- ations of mathematics, and focuses on how different foundational frameworks are related. The expression `foundations of mathematics' suggests often a reductionist approach in which some concepts are seen as fundamental and all other mathem- atical concepts ought to be defined in their terms. Here, however, we consider foundations in the sense of an approach and lan- guage for the development of mathematics. From a pragmatic viewpoint, different languages and concepts are suited to study and be applied to different areas of mathematics. The three main options for developing the foundations of mathematics are set theory, category theory and type theory, with a lot of diversity within each one. Other foundational theories are systems of arithmetic used in reverse mathematics [Sim09] and untyped formal systems, for example Feferman's system of explicit mathematics [Fef75]. When using the language and techniques of set theory, category theory or type theory, the mathematics developed there has distinct flavours, different ideas come into play and different questions arise naturally. In other words, the mathemat- ics we do and the way we do it depend on the foundational framework. One can argue that any single perspective has a blind spot, from which comes the import- ance of having available different foundational frameworks, and to understand their relationships. Despite the diversity of different foundational theories nearly all of what is used by the working mathematician can be developed in these different frameworks and to a large extent also in subsystems of second order arithmetic. This underlines the interconnectedness of these quite different foundational frameworks. In particular there are various ways to relate set theory, category theory and type theory with each other. See [Fef88] for the foundational and conceptual implications of inter- preting a theory into another one. See also [MP02] and [Awo11] for more details on relating set theory, category theory and type theory. 1 2 INTRODUCTION In this thesis we interpret constructive set theories into type theories taking inspiration from concepts and ideas from homotopy type theory. We are also in- terested in understanding how these different interpretations relate to each other. Constructive set theories and type theories Here we give some context and an introduction to the theories we consider in this thesis. Set theories. The set theories we consider are Myhill's constructive set theory CST [Myh75] and Aczel's Constructive Zermelo-Fraenkel CZF [Acz78]. We also consider for convenience the theory that we call Basic Constructive Set Theory, BCS for short, which comprises of the axioms shared by both CST and CZF. See appendix A for their axioms. The only difference between CST and CZF is that the Replacement and Exponentiation axioms of CST are replaced in CZF by the Strong Collection and Subset Collection axioms, which are stronger (see [AR10, Section 5.1]). The theories CST and CZF are constructive at two levels: in the sense that the underlying logic is intuitionistic rather than classical, and also because they are predicative theories. At the first level, one can see them intuitively as the result of the process that starts from ZFC and either removes or reformulates the axioms that are incompatible with intuitionistic logic. The axiom of choice is removed and the Foundation axiom is reformulated as the 2-induction axiom, since they both imply the law of excluded middle. Moreover, CST and CZF are predicative, meaning that sets are conceived as being constructed from already constructed sets. Definitions of a set that use a collection to which the set belongs are prevented by further restrictions on the axioms. For example the usual definition of closure in a topological space is im- predicative: `given a subset of a space S ⊆ X, the closure S¯ is the smallest closed set containing S'.