Axiomatizing Kaufmann Models of Arithmetic in Strong Logics

19.05.2022 15:00 - 16:30

C. Switzer (U Wien)

A Kaufmann model of \(\mathsf{PA}\) is an \(\omega_1\)-like, recursively saturated, rather classless model (these terms will be defined in the talk). Such models have been an important object of study in model theory of arithmetic and its environs since the 70's. Kaufmann models are natural counterexamples to several theorems about countable models of \(\mathsf{PA}\) holding at the uncountable. Moreover they are a witness to incompactness at \(\omega_1\) similar to an Aronszajn tree. The proof that Kaufmann models exist lies along a somewhat twisted road. Kaufmann showed that there are Kaufmann models under the combinatorial principle \(\diamondsuit_{\omega_1}\) and, later, Shelah eliminated the use of \(\diamondsuit_{\omega_1}\) by appealing to a forcing absoluteness argument involving the strong logic \(L_{\omega_1, \omega}(Q)\) where \(Q\) is the quantifier “there exists uncountably many”. It remains an extremely interesting, if somewhat vague, question, attributed to Hodges, whether one can build a Kaufmann model “by hand” in \(\mathsf{ZFC}\) without appealing to generic absoluteness.

In this talk we will report on our recent progress in this area. Specifically we will consider the role that the strong logic \(L_{\omega_1, \omega}(Q)\) plays in Kaufmann models and show that the statement “Kaufmann models can be axiomatized by \(L_{\omega_1, \omega}(Q)\)” is independent of \(\mathsf{ZFC}\). Along the way we will discuss how Kaufmann models are affected by forcing and in particular show that it is independent of \(\mathsf{ZFC}\) whether or not there is a Kaufmann model which can be “killed” by forcing without collapsing \(\omega_1\).




HS 13, 2. OG., OMP 1