Lifting Results for Finite Dimensions to the Transfinite in Systems of Varieties Using Ultraproducts

Abstract

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 $\mathbf{O}\mathbf{r}\mathbf{d}$ denote the class of all ordinals. Let $⟨{\mathbf{K}}_{\alpha }:\alpha \in \mathbf{O}\mathbf{r}\mathbf{d}⟩$ be a system of varieties definable by a schema. Given any ordinal $\alpha$, we define an operator ${\mathsf{Nr}}_{\alpha }$ that acts on ${\mathbf{K}}_{\beta }$ for any $\beta >\alpha$ giving an algebra in ${\mathbf{K}}_{\alpha }$, as an abstraction of taking $\alpha$-neat reducts for cylindric algebras. We show that for any positive $k$, and any infinite ordinal $\alpha$ that $\mathbf{S}{\mathsf{Nr}}_{\alpha }{\mathbf{K}}_{\alpha +k+1}$ cannot be axiomatized by a finite schema over $\mathbf{S}{\mathsf{Nr}}_{\alpha }{\mathbf{K}}_{\alpha +k}$ 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 $L_{\omega ,\omega }$.