Destructibility and axiomatizability of Kaufmann models

Abstract

AbstractA Kaufmann model is an $$\omega _1$$ ω 1 -like, recursively saturated, rather classless model of $${{\mathsf {P}}}{{\mathsf {A}}}$$ P A (or $${{\mathsf {Z}}}{{\mathsf {F}}} $$ Z F ). Such models were constructed by Kaufmann under the combinatorial principle $$\diamondsuit _{\omega _1}$$ ♢ ω 1 and Shelah showed they exist in $$\mathsf {ZFC}$$ ZFC by an absoluteness argument. Kaufmann models are an important witness to the incompactness of $$\omega _1$$ ω 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 $$\omega _1$$ ω 1 . We show that the answer to this question is independent of $$\mathsf {ZFC}$$ ZFC and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of $$\mathsf {ZFC}$$ ZFC whether or not Kaufmann models can be axiomatized in the logic $$L_{\omega _1, \omega } (Q)$$ L ω 1 , ω ( Q ) where Q is the quantifier “there exists uncountably many”.