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Martin's Axiom, Omitting Types, and Complete Representations in Algebraic Logic

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Abstract

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; nw via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin's axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey's omitting types theorem fails for L n, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. L n has been recently (and quite extensively) studied as a many-dimensional modal logic.

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Ahmed, T.S. Martin's Axiom, Omitting Types, and Complete Representations in Algebraic Logic. Studia Logica 72, 285–309 (2002). https://doi.org/10.1023/A:1021368713305

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