An algebraic result about soft model theoretical equivalence relations with an application to H. Friedman's fourth problem

Journal of Symbolic Logic 46 (3):523-530 (1981)
We prove the following algebraic characterization of elementary equivalence: $\equiv$ restricted to countable structures of finite type is minimal among the equivalence relations, other than isomorphism, which are preserved under reduct and renaming and which have the Robinson property; the latter is a faithful adaptation for equivalence relations of the familiar model theoretical notion. We apply this result to Friedman's fourth problem by proving that if L = L ωω (Q i ) i ∈ ω 1 is an (ω 1 , ω)-compact logic satisfying both the Robinson consistency theorem on countable structures of finite type and the Löwenheim-Skolem theorem for some $\lambda for theories having ω 1 many sentences, then $\equiv_L = \equiv$ on such structures
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DOI 10.2307/2273754
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References found in this work BETA
K. Jon Barwise (1974). Axioms for Abstract Model Theory. Annals of Mathematical Logic 7 (2-3):221-265.
K. J. Barwise (1974). Axioms for Abstract Model Theory. Annals of Pure and Applied Logic 7 (2):221.

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