Model Theory for a Compact Cardinal

Abstract

We like to develop model theory for $T$, a complete theory in $\mathbb{L}_{\theta,\theta}(\tau)$ when $\theta$ is a compact cardinal. By [Sh:300a] we have bare bones stability and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) we restrict ourselves to ``$D$ a $\theta$-complete ultrafilter on $I$, probably $(I,\theta)$-regular". The basic theorems work, but can we generalize deeper parts of model theory? In particular can we generalize stability enough to generalize [Sh:c, Ch.VI]? We prove that at least we can characterize the $T$'s which are minimal under Keisler's order, i.e. such that $\{D:D$ is a regular ultrafilter on $\lambda$ and $M \models T \Rightarrow M^\lambda/D$ is $\lambda$-saturated$\}$. Further we succeed to connect our investigation with the logic $\mathbb{L}^1_{< \theta}$ introduced in [Sh:797]: two models are $\mathbb{L}^1_{< \theta}$-equivalent iff \, for some $\omega$- sequence of$\theta$-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.