A Note on Extensions of Infinitary Logic
Abstract
We show that a strong form of the so called Lindstrom's Theorem fails to generalize to extensions of Lkappa,omega and Lkappa,kappa: For weakly compact kappa there is no strongest extension of Lkappa,omega with the (kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to kappa. With an additional set-theoretic assumption, there is no strongest extension of Lkappa,kappa with the (kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to <kappa.
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