  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.
