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 s a countable recursively saturated model of PA in which N is a strong cut, then for any M 0 ≺ M there is an automorphism j of M such that the fixed point set of j is isomorphic to M 0 . o 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 e M is a countable recursively saturated model of PA in which N is a strong cut. There is a group embedding j ↦ j ˆ from Aut ( Q ) into Aut ( M ) such that for each j ∈ Aut ( Q ) that is fixed point free, j ˆ moves every undefinable element of M .