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 [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 A of subsets of the set ω of natural numbers such that the expansion NA≔(N,A)A∈A of the standard model N≔(ω,+,×) of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension of NA, there is a subset of ω∗ that is parametrically definable in but whose intersection with ω is not a member of A. 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 A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra (where is the ideal of finite sets) collapses ℵ1 when viewed as a notion of forcing.