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)
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Abstract

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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Citations of this work

Compactness, interpolation and Friedman's third problem.Daniele Mundici - 1982 - Annals of Mathematical Logic 22 (2):197.

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References found in this work

Axioms for abstract model theory.K. Jon Barwise - 1974 - Annals of Mathematical Logic 7 (2-3):221-265.
Stationary logic.Jon Barwise - 1978 - Annals of Mathematical Logic 13 (2):171.
Axioms for abstract model theory.K. J. Barwise - 1974 - Annals of Mathematical Logic 7 (2-3):221-265.
δ-Logics and generalized quantifiers.J. A. Makowsky - 1976 - Annals of Mathematical Logic 10 (2):155-192.

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