Four Problems Concerning Recursively Saturated Models of Arithmetic

Abstract

The paper presents four open problems. One concerns a possible converse to Tarski's undefinability of truth theorem, and is of a general character. The other three are more specific. The questions are about some special $\omega_1$-like models, initial segments of countable recursively saturated models of PA, and about extendability of automorphisms. In each case a partial answer is given. All partial solutions are based on applications of inductive satisfaction classes.