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

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n ≤ w 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 Ln, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. Ln has been recently (and quite extensively) studied as a many-dimensional modal logic.

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