Submission date: 25 July 2021

Abstract:

We study notions of genericity in models of 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 PA which satisfy the axiomatization given by Chatzidakis and Pillay; we refer to these subsets in models of 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.

Mathematics Subject Classification: 03C62 (Primary), 03H15

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Full text arXiv 2107.11867: pdf, ps.