Abstract

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of L κ ω and L κ κ : For weakly compact κ there is no strongest extension of L κ ω with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of L κ κ with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to <κ