Automorphisms of Countable Short Recursively Saturated Models of PA

Abstract

A model of Peano Arithmetic is short recursively saturated if it realizes all its bounded finitely realized recursive types. Short recursively saturated models of $\mathrm{PA}$ are exactly the elementary initial segments of recursively saturated models of $\mathrm{PA}$. In this paper, we survey and prove results on short recursively saturated models of $\mathrm{PA}$ and their automorphisms. In particular, we investigate a certain subgroup of the automorphism group of such models. This subgroup, denoted $G{|}_{M(a)}$, contains all the automorphisms of a countable short recursively saturated model of $\mathrm{PA}$ which can be extended to an automorphism of the countable recursively saturated elementary end extension of the model.