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 . In particular, we use this method to prove Theorem A below, which confirms a long-standing conjecture of James Schmerl.
Theorem A
Ifis a countable recursively saturated model ofin whichis a strong cut, then for anythere is an automorphism j ofsuch that the fixed point set ofis isomorphic to.
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 is the automorphism group of the structure , and is the ordered set of rationals).
Theorem B
Supposeis a countable recursively saturated model ofin whichis a strong cut. There is a group embeddingfromintosuch that for eachthat is fixed point free,moves every undefinable element of.