Characterization of ${\mathcal{L}}_{\kappa }^1$

Abstract

The logic ${\mathcal{L}}_{\kappa }^1$ was introduced by Shelah in [3]. In [4], he proved that for a strongly compact cardinal $\kappa$, it admits the following algebraic characterization: two structures are ${\mathcal{L}}_{\kappa }^1$-equivalent if and only if they have isomorphic iterated ultrapowers via $\kappa$-complete ultrafilters. We give a presentation of the logic ${\mathcal{L}}_{\kappa }^1$ and a simplified and slightly modified proof of this result.