CP-generic expansions of models of Peano Arithmetic

TL;DR

This paper explores generic expansions of models of Peano Arithmetic using model-theoretic and forcing techniques, introducing CP-genericity and strong CP-genericity, and constructing models with various definability properties.

Contribution

It introduces the concepts of CP-genericity and strong CP-genericity in models of PA and applies arithmetic Cohen forcing to construct models with specific definability characteristics.

Findings

Constructed CP-generic models with all elements definable.

Built models where definable closure remains unchanged.

Analyzed properties of models satisfying CP-genericity axioms.

Abstract

We study notions of genericity in models of $\mathsf{PA}$, inspired by lines of inquiry initiated by Chatzidakis and Pillay and continued by Dolich, Miller and Steinhorn in general model-theoretic contexts. These papers studied the theories obtained by adding a "random" predicate to a class of structures. Chatzidakis and Pillay axiomatized the theories obtained in this way. In this article, we look at the subsets of models of $\mathsf{PA}$ which satisfy the axiomatization given by Chatzidakis and Pillay; we refer to these subsets in models of $\mathsf{PA}$ as CP-generics. We study a more natural property, called strong CP-genericity, which implies CP-genericity. We use an arithmetic version of Cohen forcing to construct (strong) CP-generics with various properties, including ones in which every element of the model is definable in the expansion, and, on the other extreme, ones in which the definable closure relation is unchanged.