Nonstandard definability

We investigate the notion of definability with respect to a full satisfaction class σ for a model M of Peano arithmetic. It is shown that the σ-definable subsets of M always include a class which provides a satisfaction definition for standard formulas. Such a class is necessarily proper, therefore there exist recursively saturated models with no full satisfaction classes. Nonstandard extensions of overspill and recursive saturation are utilized in developing a criterion for nonstandard definability. Finally, these techniques yield some information concerning the extendibility of full satisfaction classes from one model to another.