Truth in generic cuts

In an earlier paper (MLQ 54, 128–144) the first author initiated the study of generic cuts of a model of Peano Arithmetic relative to a notion of an indicator in the model. This paper extends that work. We generalise the idea of indicator to a related neighbourhood system; this allows the theory to be extended to one that includes the case of elementary cuts. Most results transfer to this more general context, and in particular we obtain the idea of a generic cut relative to a neighbourhood system, which is studied in more detail. The main new result on generic cuts presented here is a description of truth in the structure (M, I), where I is a generic cut of a model M of Peano Arithmetic. The special case of elementary generic cuts provides a partial answer to a question of Kossak (Notre Dame J. Formal Logic 36, 519–530).