A standard model of Peano arithmetic with no conservative elementary extension

The principal result of this paper answers a long-standing question in the model theory of arithmetic [KS, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion ΩA := (ω, +, ·, X)X∈A of the standard model of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension ΩA = (ω ∗, · · ·) of ΩA, there is a subset of ω∗ that is parametrically definable in ΩA but whose intersection with ω is not a member of A. Inspired by a recent question of Gitman and Hamkins, we also show that the aforementioned family A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra A/FIN (where FIN is the ideal of finite sets) collapses א1 when viewed as a notion of forcing.