Large Sets in Constructive Set Theory Albert Zieger Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds School of Mathematics December 2014 ii 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 acknowledgement. c 2014 The University of Leeds and Albert Ziegler The right of Albert Ziegler to be identified as Author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. iii Acknowledgements I would like to thank my supervisor Professor Michael Rathjen for sharing his vast ex- perience and for providing valuable guidance through both the mathematical and the organisational challenges of life as a PhD student. I am grateful to the Leeds Logic Group for their continued support and stimulating dis- cussions, for organising interesting talks and providing such a friendly environment. As a fellow of the network “From Mathematical Logic to Applications” (MALOA), I would also like to thank the European Union’s Marie Curie Initial Training Networks programme for their funding and training. I would also like to express my gratitude to my family for their love and support. iv Abstract This thesis presents an investigation into large sets and large set axioms in the context of the constructive set theory CZF. We determine the structure of large sets by classifying their von Neumann stages and use a new modified cumulative hierarchy to characterise their arrangement in the set theoretic universe. We prove that large set axioms have good metamathematical prop- erties, including absoluteness for the common relative model constructions of CZF and a preservation of the witness existence properties CZF enjoys. Furthermore, we use realizability to establish new results about the relative consistency of a plurality of inac- cessibles versus the existence of just one inaccessible. Developing a constructive theory of clubs, we present a characterisation theorem for Mahlo sets connecting classical and constructive approaches to Mahloness and determine the amount of induction contained in the assertion of a Mahlo set. We then present a characterisation theorem for 2-strong sets which proves them to be equivalent to a logically simpler concept. We also investigate several topics connected to elementary embeddings of the set theo- retic universe into a transitive class model of CZF, where considering different equiva- lent classical formulations results in a rich and interconnected spectrum of measurability for the constructive case. We pay particular attention to the question of cofinality of elementary embeddings, achieving both very strong cofinality properties in the case of Reinhardt embeddings and constructing models of the failure of cofinality in the case of ordinary measurable embeddings, some of which require only surprisingly low con- ditions. We close with an investigation of constructive principles incompatible with elementary embeddings. v To Annika and Fiona vi vii Contents Acknowledgements . iii Abstract . iv Dedication . .v Contents . vi List of figures . xii List of tables . xiv 1 Introduction 1 2 Preliminaries 7 2.1 Notations and Conventions . .7 2.2 CZF, Similar Constructive Set Theories and Related Axioms . 12 2.3 The Concepts of Largeness under Consideration . 20 2.3.1 Tiny Large Sets . 21 2.3.2 Small Large Sets . 24 2.3.3 Large Large Sets . 28 CONTENTS viii 3 What do Large Sets look like? 35 3.1 Introduction and Results . 35 3.2 Proof of Theorem 3.2 . 36 3.3 Why 3? . 38 3.4 A much stronger Classification Theorem in case of Subcountability . 39 4 How Constructive are Large Set Axioms? 43 4.1 Models of Constructive Set Theory . 44 4.1.1 Applicative Topologies . 45 4.1.2 Models of Set Theory Based on Applicative Topologies . 48 4.1.3 The Consistency of Tiny Large Set Axioms . 50 4.1.4 The Absoluteness of Inaccessibility . 56 4.1.5 The Consistency of Small Large Set Axioms . 64 4.1.6 The Consistency of Large Large Set Axioms . 70 4.2 The Existence of Witnesses . 77 4.2.1 Realizability with Truth . 78 4.2.2 Absoluteness Proofs . 83 4.2.3 Realizing Elementary Embeddings . 92 4.2.4 Theories with Good Properties . 95 5 How are Large Sets arranged in the Universe? 97 5.1 Ordinals and the Cumulative Hierarchy . 99 5.2 The Modified Hierarchy . 102 5.3 The Modified Hierarchy as a Hierarchy of Sets . 106 CONTENTS ix 5.4 Interaction with Large Sets . 109 5.5 Further Applications of the Methods Developed in this Chapter . 114 6 How many Large Sets are there? 117 6.1 A Realizability Model with Inaccessible pca . 120 6.2 A second Inaccessible . 124 6.3 A third Inaccessible . 130 6.4 Another ! Inaccessibles . 130 6.5 A Proper Class of Inaccessibles . 132 6.6 Proper Classes and Inexhaustible Classes . 134 6.7 An Inexhaustible Class of Inaccessibles . 136 7 A Closer Look into Mahlo Sets 139 7.1 A Constructive Rendering of Clubs . 140 7.1.1 Preliminaries and Definitions . 140 7.1.2 Limits of Clubs . 145 7.1.3 Intersections of Clubs . 149 7.1.4 An Ordinalless Approach . 152 7.2 Classical and Constructive Mahloness . 158 7.2.1 Characterising Mahlo Sets using Dependent Choice . 158 7.2.2 A Variation without Choice . 161 7.2.3 A Club Based Characterisation for Axiom M . 163 7.3 How much Induction is contained in Mahloness? . 164 7.3.1 Unleashing M . 166 CONTENTS x 7.3.2 Nondeterministic Inductive Definitions equivalent to M . 170 8 Expressing Weak Compactness 175 8.1 Different Renderings of the same Classical Concept . 175 8.2 A Simpler Characterisation of 2-Strong Sets . 179 9 Elementary Embeddings 187 9.1 How Inaccessible must the Critical Point of a Measurable be? . 188 9.1.1 Improving on Transitivity . 189 9.1.2 IEA for Critical Points with more Traction . 192 9.1.3 The Modified von Neumann Hierarchy for more Closure . 197 9.2 The Strength of Restricted Elementary Embeddings . 201 9.2.1 Inaccessibility and Mahloness . 202 9.2.2 Beyond Mahlo . 206 9.3 Cofinality of Elementary Embeddings . 210 9.3.1 The Constructive Case . 212 9.3.2 Approximating Subset Relations . 213 9.3.3 Truth Values and Elementary Embeddings . 219 9.3.4 Subset Relations and Elementary Embeddings . 222 9.4 A Model with a Map j : V ! M ..................... 225 9.4.1 The pca . 226 9.4.2 The Realizability Model . 231 9.4.3 Critical Points and Cofinality in the Model . 243 9.4.4 The Limits of the Methods used in this Chapter . 249 CONTENTS xi 9.5 Stronger Assumptions yield a Fully Elementary Embedding refuting Co- finality . 252 9.5.1 What is and is not realized to be in M .............. 253 9.6 Hitting the Ceiling . 259 Bibliography 263 LIST OF FIGURES xii xiii List of figures 5.1 An Informal Map of the Universe . 98 9.1 Closure Properties of Critical Points . 189 LIST OF TABLES xiv xv List of tables 2.1 Orders of Precedence for Logical Symbols . .8 2.2 Atomic Formulae containing Class Terms . 10 2.3 Defined Terms . 11 2.4 Types of Relations . 12 LIST OF TABLES xvi 1 Chapter 1 Introduction Large cardinals play a pivotal role in classical set theory, both as useful tools to be applied in other set theoretic pursuits and as worthy objects of study in their own right, and even as ways to bolster ZFC’s strength in order to handle mathematical challenges outside of set theory (e.g. by supplying Grothendieck universes). There is little reason to believe that in the context of constructive set theory, similar axioms would prove less fruitful. On the contrary, due to the more modest strength claimed by predicative systems of constructive set theory such as CZF, which has achieved a prominent role as foundation for constructive mathematics since its introduction in [Acz78], it seems that mathematicians working in these systems might be forced to revert to an axiom of infinity more often than their classical colleagues if they want to achieve general results. Thus the advantage of adopting axioms of largeness might be even greater than in classical set theory. And indeed in the context of CZF, axioms like REA (which can be considered to be a principle of largeness) are commonly used e.g. in the development of formal topology or for general induction. Furthermore, systems which strive to preserve some level of predicativity like CZF present avenues of investigation which are closed for systems of as yet unmanageable strength like ZFC. Most prominently, the much praised linearity of consistency strengths of different large cardinal axioms becomes a quantifiable commodity in constructive set 1 INTRODUCTION 2 theory in the sense that the proof theoretic ordinal can be exactly determined for the sys- tem of CZF enhanced by large set axioms. Indeed, the research connected to constructive analoga for large cardinals in the context of CZF has in large part been focussed on their ordinal analysis (see e.g. [Rat98, Gib02, CR02, GRT05]). In contrast, the theory of what could roughly be called the more set theoretic properties and consequences of principles of largeness has perhaps not been quite as thoroughly developed (at least above the level of what we will call “tiny” large sets, i.e. beyond reg- ularity and its variants), and this thesis aims at making some contributions to remedy this situation.