A note on extensions of infinitary logic

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ: For weakly compact κ there is no strongest extension of Lκω 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 Lκκ with the (κ, κ)-compactness property and the LöwenheimSkolem theorem down to < κ. By a well-known theorem of Lindström [4], first order logic Lωω is the strongest logic which satisifies the compactness theorem and the downward Löwenheim-Skolem theorem. For weakly compact κ, the infinitary logic Lκω satisfies both the (κ, κ)-compactness property and the Löwenheim-Skolem theorem down to κ. In [1] Jon Barwise pointed out that Lκω is not maximal with respect to these properties, and asked what is the strongest logic based on a weakly compact cardinal κ which still satisfies the (κ, κ)-compactness property and some other natural conditions suggested by κ. We prove (Corollary 5) that for weakly compact κ there is no strongest extension of Lκω with We are indebted to Lauri Hella, Tapani Hyttinen and Kerkko Luosto for useful suggestions. Research partially supported by the United States-Israel Binational Science Foundation. Publication number [ShVa:726] Research partially supported by grant 40734 of the Academy of Finland.