The Keisler Order in Continuous Logic

Abstract

We extend the Keisler order to continuous first-order theories. In the process, we show that if F is a $\lambda$-regular filter on I, and $\langle {\mathcal {M}}_i\rangle _{i \in I}$, $\langle {\mathcal {N}}_i\rangle _{i \in I}$ are sequences of continuous structures in the same language such that $\prod _F {\mathcal {M}}_i$ and $\prod _F {\mathcal {N}}_i$ have the same continuous first-order theory, then the classical structures corresponding to $\prod _F {\mathcal {M}}_i$ and $\prod _F {\mathcal {N}}_i$ satisfy the same sentences of ${\mathcal {L}_{\infty , \lambda ^+}}$ of alternating quantifier rank at most $(\lambda )$.