Ali Enayat

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 AIf is a countable recursively saturated model of in which is a strong cut, then for any there is an automorphism j of such that the fixed point set of j is 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 .Theorem BSuppose is a countable recursively saturated model of in which is a strong cut. There is a group embedding from into such that for each that is fixed point free, moves every undefinable element of

The principal result of this paper answers a long-standing question in the model theory of arithmetic [R. Kossak, J. Schmerl, The Structure of Models of Peano Arithmetic, Oxford University Press, 2006, Question 7] by showing that there exists an uncountable arithmetically closed family of subsets of the set ω of natural numbers such that the expansion of the standard model of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension of , there is a subset of ω* that is parametrically definable in but whose intersection with ω is not a member of . We also establish other results that highlight the role of countability in the model theory of arithmetic. Inspired by a recent question of Gitman and Hamkins, we furthermore show that the aforementioned family can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra collapses 1 when viewed as a notion of forcing