Initial segments and end-extensions of models of arithmetic

Abstract

This thesis is organized into two independent parts. In the first part, we extend the recent work on generic cuts by Kaye and the author. The focus here is the properties of the pairs (M, I) where I is a generic cut of a model M. Amongst other results, we characterize the theory of such pairs, and prove that they are existentially closed in a natural category. In the second part, we construct end-extensions of models of arithmetic that are at least as strong as ATR$_0$. Two new constructions are presented. The first one uses a variant of Fodor’s Lemma in ATR$_0$ to build an internally rather classless model. The second one uses some weak versions of the Galvin–Prikry Theorem in adjoining an ideal set to a model of second-order arithmetic