Destructibility and axiomatizability of Kaufmann models

A Kaufmann model is an ${\omega }_1$ -like, recursively saturated, rather classless model of $PA$ (or $ZF$ ). Such models were constructed by Kaufmann under the combinatorial principle ${♢}_{{\omega }_1}$ and Shelah showed they exist in $\mathrm{ZFC}$ by an absoluteness argument. Kaufmann models are an important witness to the incompactness of ${\omega }_1$ 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 ${\omega }_1$ . We show that the answer to this question is independent of $\mathrm{ZFC}$ and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of $\mathrm{ZFC}$ whether or not Kaufmann models can be axiomatized in the logic $L_{{\omega }_1,\omega }\left(Q\right)$ where Q is the quantifier "there exists uncountably many".