Automorphisms of models of arithmetic: A unified view

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Abstract

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 M0Mthere 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 jjˆ from Aut(Q) into Aut(M) such that for each jAut(Q)that is fixed point free, jˆ moves every undefinable element of M .

Keywords

Automorphism group
Recursive saturation
Peano arithmetic
Strong cut
Iterated ultrapower

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