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

Let$\left\langle {{W_n}:n \in \omega } \right\rangle$be a canonical enumeration of recursively enumerable sets, and supposeTis a recursively enumerable extension of PA (Peano Arithmetic) in the same language. Woodin (2011) showed that there exists an index$e \in \omega$(that depends onT) with the property that if${\cal M}$is a countable model ofTand for some${\cal M}$-finite sets,${\cal M}$satisfies${W_e} \subseteq s$, then${\cal M}$has an end extension${\cal N}$that satisfiesT+We=s.Here we generalize Woodin’s theorem to all recursively enumerable extensionsTof the fragment${{\rm{I}\rm{\Sigma }}_1}$of PA, and remove the countability restriction on${\cal M}$whenTextends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.