Geodesic
Lifting Results for Finite Dimensions to the Transfinite in Systems of Varieties Using Ultraproducts
Bulletin of the Section of Logic, Tome 53 (2024) no. 2, pp. 145-154.

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We redefine a system of varieties definable by a schema of equations to include finite dimensions. Then we present a technique using ultraproducts enabling one to lift results proved for every finite dimension to the transfinite. Let Ord denote the class of all ordinals. Let 〈𝐊_α: α∈ Ord〉 be a system of varieties definable by a schema. Given any ordinal α, we define an operator 𝖭𝗋_α that acts on 𝐊_β for any β gt;α giving an algebra in 𝐊_α, as an abstraction of taking α-neat reducts for cylindric algebras. We show that for any positive k, and any infinite ordinal α that 𝐒𝖭𝗋_α𝐊_α+k+1 cannot be axiomatized by a finite schema over 𝐒𝖭𝗋_α𝐊_α+k given that the result is valid for all finite dimensions greater than some fixed finite ordinal. We apply our results to cylindric algebras and Halmos quasipolyadic algebras with equality. As an application to our algebraic result we obtain a strong incompleteness theorem (in the sense that validitities are not captured by finitary Hilbert style axiomatizations) for an algebraizable extension of L_ω,ω.
Keywords: algebraic logic, systems of varieties, ultraproducts, non-finite axiomaitizability 
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Sayed Ahmed, Tarek. Lifting Results for Finite Dimensions to the Transfinite in Systems of Varieties Using Ultraproducts. Bulletin of the Section of Logic, Tome 53 (2024) no. 2, pp. 145-154. http://geodesic.mathdoc.fr/item/BSL_2024_53_2_a0/

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