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The Structure of Models of Second-Order Set Theories

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The Structure of Models of Second-Order Set Theories The Structure of Models of Second-order Set Theories by Kameryn J Williams arXiv:1804.09526v1 [math.LO] 25 Apr 2018 A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York. 2018 ii c 2018 Kameryn J Williams All Rights Reserved iii The Structure of Models of Second-order Set Theories by Kameryn J Williams This manuscript has been read and accepted by the Graduate Faculty in Mathematics in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy. Professor Joel David Hamkins Date Chair of Examining Committee Professor Ara Basmajian Date Executive Officer Professor Joel David Hamkins Professor Arthur Apter Professor Gunter Fuchs Supervisory Committee The City University of New York iv Abstract The Structure of Models of Second-order Set Theories by Kameryn J Williams Advisor: Professor Joel David Hamkins This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, or- ganized by chapter. First, I investigate the poset of T -realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from KM to weaker theories. They showed that every model of KM plus the Class Collection schema “unrolls” to a model of ZFC− with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as GBC+ETR. I also show that being T -realizable goes down to submodels for a broad selection of second-order set theories T . Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have KM 1 CA least models. I show that strong theories—such as or Π1- —do not have least tran- sitive models while weaker theories—from GBC to GBC + ETROrd—do have least transitive models. Dedication For Margaret and Martin. v Acknowledgments First and foremost I must thank Joel. You have been a fantastic advisor throughout my studies and there is no doubt that my growth as a mathematician owes much to your support and guidance. Thank you for the many fruitful conversations, for thoroughly reading through this text, and for the myriad helpful suggestions along the way. Thank you Arthur and Gunter for serving on my dissertation committee and for reading and giving me comments on this document. Thank you Roman for being on my committee in spirit and for organizing the models of arithmetic group at CUNY. Thank you Kaethe, Miha, Corey, Alex, Ryan, and Eoin. It was great to have other students interested in set theory and to have a student-run seminar in the subject for all of my years at CUNY. Thank you Vika for many conversations and for your hard work in making things run smoothly for the set theory group. And thank you to all the others who make CUNY’s logic community so vibrant and such a productive place to be. Thank you Herbie. You are my best friend and I love you. Thank you for moving across the country with me and for supporting and putting up with me through the joys and trials of graduate school. And finally, thank you mom, dad, Chris, Kalen, Christian, and Lexi. vi Contents Contents vii List of Figures ix 0 Introduction 1 1 A first look at models of second-order set theories 5 1.1 DramatisPersonae ................................ 6 1.1.1 Second-ordersettheories. .. .. 8 1.1.2 DoingwithoutPowerset ......................... 15 1.1.3 Modelsofsecond-ordersettheory . 16 1.2 Verifyingtheaxioms ............................... 19 1.3 Preservingtheaxioms .............................. 24 1.4 Somebasicconstructions. .. .. 31 1.5 GBC-realizationsofacountablemodel . 41 2 Many constructions: unrollings, cutting offs, and second-order L 61 2.1 The unrolling construction . 66 2.1.1 Unrolling in GBC + ETR ......................... 82 2.1.2 How hard is it to unroll? . 88 2.2 Thecuttingoffconstruction . 91 vii CONTENTS viii 2.3 The constructible universe in the classes . .. 96 2.4 The smallest heights of transitive models and β-models ............ 107 3 Truth and transfinite recursion 112 3.1 Truthanditeratedtruth ............................. 115 3.2 Iterated truth and constructibility . 121 3.3 Separating levels of ETR ............................. 126 3.4 Adetourthroughthefiniterealm . 131 3.5 Non-elementarytransfiniterecursions . .... 140 4 Least models 151 4.1 β-modelsandwell-foundedclasses. 157 4.2 Admissible covers and nonstandard compactness arguments . ........ 159 4.3 Strongtheories .................................. 168 4.4 Mediumtheories ................................. 179 4.5 Weaktheories................................... 189 4.6 Coda: theanalogytosecond-orderarithmetic . .... 191 Bibliography 195 List of Figures 1.1 Some classes of models of second-order set theory. ....... 18 2.1 A membership code representing the set a = {0, 2}............... 67 2.2 To construct a membership code for X we remove all edges ending at the S top node and add edges (nZ , tX ) for all nZ ⊸ nY ⊸ tX . ........... 72 2.3 A partial picture of W . The nodes x, y are from elts A with x<y. Below x and y the membership code W looks like A. .................. 74 2.4 A partial picture of F ⋆............................... 76 2.5 The least heights of transitive and β-models of fragments of KM. The ordinals in the picture are ordered by ≪.......................... 108 3.1 Transfinite recursion, from GBC to KM. Ordered by consistency strength. 114 3.2 The tree of inductive iterated full satisfaction classes over a model of arith- metic. The node at depth m has continuum many extensions to leaf nodes of depth k....................................... 141 4.1 An illustration of Friedman’s theorem. The dashed line represents that N is ill-founded at A................................... 160 4.2 A picture of the desired generalization of Friedman’s theorem to the ill- foundedrealm. .................................. 166 ix LIST OF FIGURES x N W 4.3 Friedman’s theorem gives N |= S with Vκ = M = Vκ and N ill-founded at A.170 4.4 The (GBc + ETR)-realizations for M form a disjoint collection of cones, each onewithabasiselementatthebottom. 182 4.5 Arithmeticversussettheory.. 191 Chapter 0 Introduction Eine Vielheit kann n¨amlich so beschaffen sein, daß die Annahme eines “Zusammenseins” aller iherer Elemente auf einen Widerspruch f¨uhrt, so daß es unm¨oglich ist, die Vielheit als eine Einheit, als “ein fertiges Ding” aufzufassen. Solche Vielheiten nenne ich absolut unendliche oder inconsistente Vielheiten. Georg Cantor The distinction between set and class can be traced back to Cantor. He distinguished sets from those multiplicities he termed absolutely infinite or inconsistent. As the name suggests, inconsistent multiplicities are those which lead to contradiction if taken as a set. For example, if we assume that the collection of ordinals is a set then we can derive the Burali–Forti paradox. From a more modern perspective based upon the iterative conception of set we have a clear distinction between sets and classes. A collection is a set if it appears at some stage α in the cumulative hierarchy, while a (proper) class consists of elements unbounded in rank. So sets can be elements of other collections, while classes can never be elements. 1 CHAPTER 0. INTRODUCTION 2 In ordinary set theoretic practice in the early twenty-first century classes are treated as mere syntactic sugar; for example, x ∈ Ord is an abbreviation for the formula expressing that x is a transitive set linearly ordered by ∈. But set theory can also be formalized with classes as actual objects, rather than relegating them to a metatheoretic role. The first axiomatization of second-order set theory—set theory with both sets and classes, sometimes called class theory—is due to von Neumann [Neu25]. His system survives into the modern day as GBC, allowing only predicative definitions of classes.1 But GBC is not the only well-studied axiomatization of second-order set theory. The other major axiom- atization KM, which allows impredicative comprehension, was independently proposed by multiple logicians—among them Morse, Quine, and Tarski. See the appendix to [Kel75] for a popularization of this system. In the decades following von Neumann’s axiomatization, second-order axiomatizations of set theory saw significant use among set theorists. Perhaps most notably, G¨odel’s original presentation [G¨od38] of his relative consistency proof for the axiom of choice was in terms of von Neumann’s system. But over time the use of second-order systems waned, with the first-order system ZFC becoming the de facto standard. In recent years, however, second-order set theory has enjoyed a revived interest. Several mathematicians independently arrived at second-order set theory as the natural arena in which to pursue certain projects. It has seen use in work on the foundations of class forcing [Ant15; HKLNS16; HKSb; HKSa; GHHSW17], hyperclass forcing [AF], formalizing the inner model hypothesis [ABF], determinacy for class games [GH17; Hac16], and in truth theoretic work [Fuj12]. One fact that has emerged is that GBC and KM are not the only interesting second-order 1But note that the contemporary GBC is quite different from von Neumann’s original axiomatization.
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