Destructibility and axiomatizability of Kaufmann models.

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