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. (shrink)

We prove that for every Scott set S there are S-saturated real closed fields and S-saturated models of Presburger arithmetic.

During the Notre Dame workshop on Vaught's Conjecture, Hjorth and Kechris asked which Borel equivalence relations can arise as the isomorphism relation for countable models of a first-order theory. In particular, they asked if the isomorphism relation can be essentially countable but not tame. We show this is not possible if the theory has uncountably many types.