Destructibility and axiomatizability of Kaufmann models

A Kaufmann model is an [Formula: see text] -like, recursively saturated, rather classless model of [Formula: see text] (or [Formula: see text] ). Such models were constructed by Kaufmann under the combinatorial principle [Formula: see text] and Shelah showed they exist in [Formula: see text] by an absoluteness argument. Kaufmann models are an important witness to the incompactness of [Formula: see text] similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly naïve question of whether such a model can be “killed” by forcing without collapsing [Formula: see text] . We show that the answer to this question is independent of [Formula: see text] and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of [Formula: see text] whether or not Kaufmann models can be axiomatized in the logic [Formula: see text] where Q is the quantifier “there exists uncountably many”.