Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.For any structure, let Aut() be its automorphism group. There are groups which are not isomorphic to any model= (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut((, <)), must be (...) torsion-free. However, as will be proved in this paper,if(A, <)is a linearly ordered set and G is a subgroup of Aut((A, <)),then there are modelsofPAsuch that Aut() ≅G.Ifis a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets ofAbe the basic open subgroups. If′ is an expansion of, then Aut(′) is a closed subgroup of Aut(). Conversely, for any closed subgroupG≤ Aut() there is an expansion′ ofsuch that Aut(′) =G. Thus, ifis a model of PA, then Aut() is not only a subgroup of Aut((N, <)), but it is even aclosedsubgroup of Aut((N, ′)).There is a characterization, due to Cohn [2] and to Conrad [3], of those groupsGwhich are isomorphic to closed subgroups of automorphism groups of linearly ordered sets. (shrink)

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.

We discuss Robinson's model theoretic proof of Tarski's theorem on undefinability of truth. We present two other “diagonal-free” proofs of Tarski's theorem, and we compare undefinability of truth to other forms of undefinability in nonstandard models of arithmetic.

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.

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