**July 8**

**Corey Switzer**,
CUNY

**Axiomatizing Kaufmann models in strong logics**

A Kaufmann model is an $\omega_1$-like, recursively saturated, rather classless model of PA. Such models were constructed by Kaufmann under the $\diamondsuit$ assumption and then shown to exist in ZFC by Shelah using an absoluteness argument involving the logic $L_{\omega_1, \omega}(Q)$ where $Q$ is the quantifier 'there exists uncountably many…'. It remains an intriguing, if vague, open problem whether one can construct a Kaufmann model in ZFC 'by hand' i.e. without appealing to some form of absoluteness or other very non-constructive methods. In this talk I consider the related problem of axiomatizing Kaufmann models in $L_{\omega_1, \omega}(Q)$ and show that this is independent of ZFC. Along the way we'll see that it is also independent of ZFC whether there is an $\omega_1$-preserving forcing notion adding a truth predicate to a Kaufmann model.