The paper presents four open problems concerning recursively saturated models of Peano Arithmetic. One problems concerns a possible converse to Tarski's undefinability of truth theorem. The other concern elementary cuts in countable recursively saturated models, extending automorphisms of countable recursively saturated models, and Jonsson models of PA. Some partial answers are given.

The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructures.

A model of Peano Arithmetic is short recursively saturated if it realizes all its bounded finitely realized recursive types. Short recursively saturated models of $\PA$ are exactly the elementary initial segments of recursively saturated models of $\PA$. In this paper, we survey and prove results on short recursively saturated models of $\PA$ 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 which can be extended to an automorphism of the countable recursively saturated elementary end extension of the model. (shrink)

In this paper we investigate the properties of automorphism groups of countable short recursively saturated models of arithmetic. In particular, we show that Kaye's Theorem concerning the closed normal subgroups of automorphism groups of countable recursively saturated models of arithmetic applies to automorphism groups of countable short recursively saturated models as well. That is, the closed normal subgroups of the automorphism group of a countable short recursively saturated model of PA are exactly the stabilizers of the invariant cuts of the (...) model which are closed under exponentiation. This Galois correspondence is used to show that there are countable short recursively saturated models of arithmetic whose automorphism groups are not isomorphic as topological groups. Moreover, we show that the automorphism groups of countable short arithmetically saturated models of PA are not topologically isomorphic to the automorphism groups of countable short recursively saturated models of PA which are not short arithmetically saturated. (shrink)

In this paper we study the automorphism groups of models of Peano Arithmetic. Kossak, Kotlarski, and Schmerl [9] shows that the stabilizer of an unbounded element a of a countable recursively saturated model of Peano Arithmetic M is a maximal subgroup of Aut if and only if the type of a is selective. We extend this result by showing that if M is a countable arithmetically saturated model of Peano Arithmetic, Ω ⊂ M is a very good interstice, and a (...) ∈ Ω, then the stabilizer of a is a maximal subgroup of Aut if and only if the type of a is selective and rational. (shrink)

We discuss automorphisms of saturated models of PA and boundedly saturated models of PA. We show that Smoryński's Lemma and Kaye's Theorem are not only true for countable recursively saturated models of PA but also true for all boundedly saturated models of PA with slight modifications.

In this paper we discuss automorphism groups of saturated models and boundedly saturated models of $\mathsf{PA}$. We show that there are saturated models of $\mathsf{PA}$ of the same cardinality with nonisomorphic automorphism groups. We then show that every saturated model of $\mathsf{PA}$ has short saturated elementary cuts with nonisomorphic automorphism groups.

The main result of this paper partially answers a question raised in about the existence of countable just recursively saturated models of Peano Arithmetic with non‐isomorphic automorphism groups. We show the existence of infinitely many countable just recursively saturated models of Peano Arithmetic such that their automorphism groups are not topologically isomorphic. We also discuss maximal open subgroups of the automorphism group of a countable arithmetically saturated model of in a very good interstice.

We give a survey of automorphisms of countable recursively saturated models of Peano Arithmetic.

Let M be a countable recursively saturated model of Th(), and let GAut(M), considered as a topological group. We examine connections between initial segments of M and subgroups of G. In particular, for each of the following classes of subgroups HG, we give characterizations of the class of terms of the topological group structure of H as a subgroup of G. (a) for some (b) for some (c) for some (d) for some (Here, M(a) denotes the smallest M containing a, (...) , , and .). (shrink)

Continuing the earlier research in [1] and [4] we work out a class of interstices in countable arithmetically saturated models of PA in which selective types are realized and a class of interstices in which 2-indiscernible types are realized.