A Note on Extensions of In nitary Logic
نویسنده
چکیده
We show that a strong form of the so called Lindstrr om's Theorem 4] fails to generalize to extensions of L ! and L : For weakly compact there is no strongest extension of L ! with the (;)-compactness property and the LL owenheim-Skolem theorem down to. With an additional set-theoretic assumption, there is no strongest extension of L with the (;)-compactness property and the LL owenheim-Skolem theorem down to <. By a well-known theorem of Lindstrr om 4], rst order logic L !! is the strongest logic which satisiies the compactness theorem and the downward LL owenheim-Skolem theorem. For weakly compact , the innnitary logic L ! satisses both the (;)-compactness property and the LL owenheim-Skolem theorem down to. In 1] Jon Barwise pointed out that L ! is not maximal with respect to these properties, and asked what is the strongest logic based on a weakly compact cardinal which still satisses the (;)-compactness We are indebted to Lauri Hella, Tapani Hyttinen and Kerkko Luosto for useful suggestions .
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