Automorphisms of models of arithmetic: A unified view

AbstractWe 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 AIf M is a countable recursively saturated model of PA in which N is a strong cut, then for any M0≺Mthere 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 BSuppose M is a countable recursively saturated model of PA in which N is a strong cut. There is a group embedding j↦jˆ from Aut(Q) into Aut(M) such that for each j∈Aut(Q)that is fixed point free, jˆ moves every undefinable element of M