Destructibility and axiomatizability of Kaufmann models

Abstract

A Kaufmann model is an \(\omega _1\)-like, recursively saturated, rather classless model of \({{\mathsf {P}}}{{\mathsf {A}}}\) (or \({{\mathsf {Z}}}{{\mathsf {F}}} \)). Such models were constructed by Kaufmann under the combinatorial principle \(\diamondsuit _{\omega _1}\) and Shelah showed they exist in \(\mathsf {ZFC}\) 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 \(\mathsf {ZFC}\) and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of \(\mathsf {ZFC}\) whether or not Kaufmann models can be axiomatized in the logic \(L_{\omega _1, \omega } (Q)\) where Q is the quantifier “there exists uncountably many”.