Class forcing and second-order arithmetic Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨atBonn vorgelegt von Regula Krapf aus Z¨urich, die Schweiz Bonn, Oktober 2016 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨atBonn 1. Gutachter: Prof. Dr. Peter Koepke 2. Gutachter: Prof. Dr. Joel David Hamkins Tag der Promotion: 12. Januar 2017 Erscheinungsjahr: 2017 Summary We provide a framework in a generalization of G¨odel-Bernays set theory for performing class forcing. The forcing theorem states that the forcing relation is a (definable) class in the ground model (definability lemma) and that every statement that holds in a class- generic extension is forced by a condition in the generic filter (truth lemma). We prove both positive and negative results concerning the forcing theorem. On the one hand, we show that the definability lemma for one atomic formula implies the forcing theorem for all formulae in the language of set theory to hold. Furthermore, we introduce several properties which entail the forcing theorem. On the other hand, we give both counterex- amples to the definability lemma and the truth lemma. In set forcing, the forcing theorem can be proved for all forcing notions by constructing a unique Boolean completion. We show that in class forcing the existence of a Boolean completion is essentially equivalent to the forcing theorem and, moreover, Boolean completions need not be unique. The notion of pretameness was introduced to characterize those forcing notions which preserve the axiom scheme of replacement. We present several new characterizations of pretameness in terms of the forcing theorem, the preservation of separation, the existence of nice names for sets of ordinals and several other properties. Moreover, for each of the aforementioned properties we provide a corresponding characterization of the Ord-chain condition. Finally, we prove two equiconsistency results which compare models of ZFC (with large cardinal properties) and models of second-order arithmetic with topological regularity properties (and determinacy hypotheses). We apply our previous results on class forcing 1 to show that many important arboreal forcing notions preserve the Π1-perfect set property over models of second-order arithmetic and also give an example of a forcing notion which 1 implies the Π1-perfect set property to fail in the generic extension. Zusammenfassung Wir f¨uhrenKlassenforcing im axiomatischen Rahmen einer Verallgemeinerung von G¨odel- Bernays-Mengenlehre ein. Das Forcing-Theorem besagt, dass die Forcingrelation eine (definierbare) Klasse im Grundmodell ist (Definierbarkeitslemma), und dass jede Aus- sage in einer generischen Erweiterung von einer Bedingung im generischen Filter erzwun- gen wird (Wahrheitslemma). Wir beweisen sowohl positive als auch negative Resultate ¨uber das Forcing-Theorem. Einerseits zeigen wir, dass das Definierbarkeitslemma f¨ur eine einzige atomare Formel reicht, um das Forcing-Theorem f¨uralle Formeln in der Sprache der Mengenlehre zeigen. Außerdem stellen wir mehrere kombinatorische Eigen- schaften von Klassenforcings vor, welche das Forcing-Theorem implizieren. Andrerseits pr¨asentieren wir Gegenbeispiele f¨urdas Definierbarkeitslemma sowie f¨urdas Wahrheit- slemma im Kontext von Klassenforcing. Im Mengenforcing ist das Forcing-Theorem eine Konsequenz der Existenz einer eindeutigen Booleschen Vervollst¨andigung. Wir zeigen, dass im Klassenforcing die Existenz einer Booleschen Vervollst¨andigung im Wesentlichen ¨aquivalent zum Forcing-Theorem ist, und dass Boolesche Vervollst¨andiungenim Allge- meinen nicht eindeutig sind. Pretameness ist eine Eigenschaft von Klassenforcings, welche definiert wurde um die Erhaltung des Ersetzungsaxioms zu charakterisieren. Wir beweisen mehrere neue Charak- terisierungen von Pretameness anhand des Forcing-Theorems, der Erhaltung des Ausson- derungsaxioms, der Existenz von Nice Names f¨urMengen von Ordinalzahlen sowie weit- eren Eigenschaften von Klassenforcings. Des Weiteren verwenden wir alle diese Eigen- schaften um die Ord-Kettenbedingung zu charakterisieren. Zu guter Letzt geben wir zwei Aquikonsistenzresultate¨ an, welche Modelle von ZFC (mit grossen Kardinalzahlen) und Modelle der zweistufigen Arithmetik mit topologis- cher Regularit¨at(und Determiniertheit) vergleichen. Wir wenden unsere Resultate ¨uber Klassenforcing an um nachzuweisen, dass zahlreiche wichtige Beispiele von Baumforcings 1 die Π1-perfekte-Teilmengeneigenschaft ¨uber Modelle der zweistufigen Arithmetik erhal- 1 ten. Andrerseits erl¨auternwir ein Beispiel eines Klassenforcings, welches die Π1-perfekte- Teilmengeneigenschaft in generischen Erweiterungen zerst¨ort. Acknowledgements On a general note, I would like to acknowledge the support which has been offered to me by the Hausdorff Center of Mathematics, University of Bonn, and the Bonn International Graduate School of Mathematics (BIGS). Firstly, I would like to express my gratitude to my advisor Prof. Dr. Peter Koepke for teaching me both in set theory and in formal mathematics as well as for the many inspiring discussions not only about my research but also about foundational and philosophical issues related to set theory and formal mathematics, and for giving me the opportunity to gain teaching experience at University of Bonn. Furthermore, I would like to thank the mathematical logic group of Bonn for their friendship, moral support and many informal coffee-break discussions which have lead us to consider many interesting open problems and fruitful research. In particular, I would like to show my gratitude to Peter Holy and Philipp Schlicht for many intriguing conversations about class forcing which have resulted in several joint papers. Many results presented in this thesis have originated in this research project. It has been a great pleasure to learn from them and work with them. They have also made many suggestions which have helped me to improve this thesis. On a further note, I would also like to thank Philipp L¨ucke and Ana Njegomir who have also worked on the first part of our project on class forcing. Moreover, Ana has been a wonderful officemate and friend and it has helped me a lot to share both the successful and frustrating moments during our common time as PhD students. Additionally, I would like to mention the further (former) members of the mathematical logic group of Bonn, Thilo Weinert, Ioanna Dimitriou and Aleksandra Kwiatkowska; I have really enjoyed to share common lunch and coffee breaks and discuss about set theory and formal mathematics, but also many other topics which are not related to mathematics. I would also like to thank Lorenz Halbeisen for his lecture and seminars during my master studies at University of Z¨urich and ETH Z¨urich and for arousing my interest in logic and set theory. It was due to him that I was motivated to apply for the PhD program at University of Bonn. Furthermore, I would like to thank my friends and fellow PhD students in Bonn for many common coffee breaks and enjoyable evenings, in particular Ana, Gennady, Michal, Lisa, Christian and Duygu. For their moral support I would also like to thank my long- term friends from Switzerland Yvonne, Karin, Anita and Eliane. Last but not least, I would like to show gratitude to my parents Johanna and Bernhard Krapf for their support and for always believing in me, as well as my brothers Tobias and Beat. A final thanks goes to my partner Claudius Ruch for the many wonderful moments that we have spent together besides my work as a PhD student. Contents 0 Introduction 1 1 A general setting for class forcing 7 1.1 Axiomatic frameworks for class forcing . .7 1.1.1 Subsystems of ZFC ...........................7 1.1.2 Second-order arithmetic . .8 1.1.3 G¨odel-Bernays set theory . .9 1.1.4 Kelley-Morse class theory . 12 1.2 Class forcing . 12 1.3 Examples . 17 1.3.1 Variants of collapse forcings . 18 1.3.2 Forcing a global well-order . 19 1.3.3 Friedman's forcing F .......................... 20 1.3.4 The class version of the L´evycollapse . 21 1.3.5 Easton forcing . 23 2 The forcing theorem 25 2.1 The forcing theorem . 25 2.1.1 Reducing the forcing theorem to atomic formulae . 26 2.1.2 The forcing theorem for infinitary formulae . 30 2.1.3 Products . 33 2.2 Pretameness . 34 2.2.1 Pretameness implies the forcing theorem . 34 2.2.2 Examples . 37 2.2.3 Two-step iterations . 39 2.3 The set decision property . 41 2.4 Approachability by projections . 42 2.5 Failures of the forcing theorem . 44 2.5.1 A failure of the definability lemma . 45 2.5.2 A failure of the truth lemma . 49 3 Boolean completions 53 3.1 (Pre-)Boolean algebras and completions . 53 3.1.1 Pre-Boolean algebras . 53 3.1.2 (Pre-)Boolean completions . 55 7 Contents 3.1.3 The separative quotient . 56 3.2 Boolean completions and the forcing theorem . 57 3.2.1 Boolean values . 57 3.3 Boolean completions in KM .......................... 61 3.4 Unions of set-sized complete subforcings . 63 3.5 Non-unique Boolean completions . 65 4 Characterizations of pretameness and the Ord-cc 69 4.1 Preliminaries . 69 4.2 The forcing theorem . 73 4.2.1 Pretameness in terms of the forcing theorem . 74 4.2.2 The maximality principle . 78 4.3 Preservation of axioms . 79 4.3.1 Failures of separation . 79 4.3.2 Separation implies replacement . 82 4.3.3 Pretameness and the preservation of axioms . 83 4.4 Boolean completions . 84 4.5 The extension maximality principle . 86 4.6 Nice forcing notions . 88 4.6.1 Basic definitions and examples . 88 4.6.2 Pretameness and nice forcing notions . 90 4.6.3 The Ord-cc and very nice forcing notions . 94 5 Second-order arithmetic, topological regularity and sharps 97 5.1 Second-order arithmetic and ZFC− ...................... 97 5.2 Inner models of ZFC and ZFC# in models of SOA .............. 104 5.2.1 Inner models of ZFC .........................