Inner Model from the Cofinality Quantifier

This thesis discusses the inner model obtained from the cofinality quantifier introduced in the paper Inner Models From Extended Logics: Part 1 by Juliette Kennedy, Menachem Magidor and Jouko Väänänen, to appear in the Journal of Mathematical Logic. The paper is a contribution to inner model theory, presenting many different inner models obtained by replacing first order logic by extended logics in the definition of the constructible hierarchy. We will focus on the model C* obtained from the logic that extends first order logic by the cofinality quantifier for \omega. The goal of this thesis is to present two major theorems of the paper and the theory that is needed to understand their proofs. The first theorem states that the Dodd-Jensen core model of V is contained in C*. The second theorem, the Main Theorem of the thesis, is a characterization of C* assuming V = L[U]. Chapters 2-5 present the theory needed to understand the proofs. Our presentation in these chapters mostly follows standard sources but we present the proofs of many lemmas in much greater detail than our source material. Chapter 2 presents the basics of iterated ultrapowers. If a model M of ZFC^- satisfies “U is a normal ultrafilter on \kappa” for some ordinal \kappa, then we can construct its ultrapower by U. We can take the ultrapower of the resulting model M1 and then continue taking ultrapowers at successor ordinals and direct limits at limit ordinals. If the constructed iterated ultrapowers M_\alpha are well-founded for all ordinals \alpha, the model M is called iterable. Chapter 3 presents L[A], the class of sets constructible relative to a set or class A. The hierarchy L_\alpha[A] is a generalization of the constructible hierachy L_\alpha. The difference is that the formulas defining the successor level L(\alpha+1)[A] can use A \ L_\alpha[A] as a unary predicate. The Main Theorem uses the model L[U], where U is a normal measure on some cardinal \kappa. Chapter 4 presents the basics of Prikry forcing, a notion of forcing defined from a measurable cardinal. The sequence of critical points of the iterable ultrapowers of L[U] generates a generic set for the Prikry forcing defined from the critical point of the \omega-th iterated ultrapower. Chapter 5 presents the theory of the Dodd-Jensen core model which is an important inner model. The core model is based on the Jensen hierarchy J_\alpha^A which produces L[A] as the union of all levels. The theory is concerned with so called premice which are levels of the J-hierarchy J_\alpha^U satisfying “U is a normal ultrafilter on \kappa” for some ordinal \kappa. A mouse is a premouse satisfying some specific properties and the core model K is the union of all mice. The last chapter presents the approach of the paper in detail. We present the definition of C(L*), the class of sets constructible using an extended logic L*, and the exact definition of C*. Then we present the proofs of the two major theorems mentioned above. The chapter naturally follows the paper but presents the proofs in greater detail and adds references to lemmas in the previous chapters that are needed for the arguments in the proofs