Automorphisms of models of arithmetic: A unified view

We develop the method of iterated ultrapower representation to provide a unified and perspicuous approach for building automorphisms of countable recursively saturated models of Peano arithmetic PA. In particular, we use this method to prove Theorem A below, which confirms a long standing conjecture of James Schmerl. Theorem A. If M is a countable recursively saturated model of PA in which N is a strong cut, then for any M0 ≺ M there is an automorphism j of M such that the fixed point set of j is isomorphic to M0. We also fine-tune a number of classical results. One of our typical results in this direction is Theorem B below, which generalizes a theorem of Kaye-Kossak-Kotlarski (in what follows Aut(X) is the automorphism group of the structure X, and Q is the ordered set of rationals). Theorem B. Suppose M is a countable recursively saturated model of PA in which N is a strong cut . There is a group embedding j 7→ ̂ from Aut(Q) into Aut(M) such that for each j ∈ Aut(Q) that is fixed point free, ̂ moves every undefinable element of M. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ 2000 Mathematics Subject Classification. Primary 03F30, 03C62, 03H15; Secondary 03C15.