A note on extensions of infinitary logic

We show that a strong form of the so called Lindström's Theorem 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.