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

## Authors

• Tarek Sayed Ahmed Cairo University, Department of Mathematics, Faculty of Science, Giza, Egypt

## Keywords:

algebraic logic, systems of varieties, ultraproducts, non-finite axiomaitizability

## 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 $$\bf Ord$$ denote the class of all ordinals. Let $$\langle \mathbf{K}_{\alpha}: \alpha\in \bf Ord\rangle$$ 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 $$L_{\omega,\omega}$$.

## References

L. Henkin, J. Monk, A. Tarski, Cylindric Algebras, Part I, vol. 64 of Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam (1971).

L. Henkin, J. Monk, A. Tarski, Cylindric Algebras, Part II, vol. 115 of Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam (1985), URL: https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/115/suppl/C

R. Hirsch, T. S. Ahmed, The neat embedding problem for algebras other than cylindric algebras and for infinite dimensions, Journal of Symbolic Logic, vol. 79(1) (2014), pp. 208–222, DOI: https://doi.org/10.1017/jsl.2013.20
Google Scholar DOI: https://doi.org/10.1017/jsl.2013.20

R. Hirsch, I. Hodkinson, Relation algebras by games, vol. 147 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam (2002), URL: https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/147/suppl/C

R. Hirsch, I. Hodkinson, R. Maddux, Relation algebra reducts of cylindric algebras and an application to proof theory, Journal of Symbolic Logic, vol. 67(1) (2002), pp. 197–213, DOI: https://doi.org/10.2178/jsl/1190150037
Google Scholar DOI: https://doi.org/10.2178/jsl/1190150037

I. Sain, R. Thompson, Strictly finite schema axiomatization of quasi-polyadic algebras, [in:] H. Andr'eka, D. Monk, I. N'emeti (eds.), Algebraic Logic, North Holland, Amsterdam (1991), pp. 539–572.

2024-02-12

## How to Cite

Sayed Ahmed, T. (2024). Lifting Results for Finite Dimensions to the Transfinite in Systems of Varieties Using Ultraproducts. Bulletin of the Section of Logic, 53(2), 145–154. https://doi.org/10.18778/0138-0680.2024.02

Article