### A Note on Extensions of Infinitary Logic

by Shelah and Vaananen. [ShVa:726]

Archive for Math Logic, 2005

We show that a strong form of the so called Lindstr{o}m's
Theorem fails to generalize to extensions of L_{kappa omega} and
L_{kappa kappa} : For weakly compact kappa there is no
strongest extension of L_{kappa omega} with the
(kappa, kappa)-compactness property and the L{o}wenheim-Skolem
theorem down to kappa . With an additional set-theoretic
assumption, there is no strongest extension of L_{kappa kappa}
with the (kappa, kappa)-compactness property and the
L{o}wenheim-Skolem theorem down to < kappa .

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