  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.
