Neutrally Expandable Models of Arithmetic

A subset of a model of ${sf PA}$ is called neutral if it does not change the $mathrm{dcl}$ relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non-existence of neutral sets in various models of ${sf PA}$. We show that cofinal extensions of prime models are neutrally expandable, and $omega_1$-like neutrally expandable models exist, while no recursively saturated model is neutrally expandable. We also show that neutrality is not a first-order property. In the last section, we study a local version of neutral expandability.