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Relation algebra reducts of cylindric algebras and an application to proof theory

Published online by Cambridge University Press:  12 March 2014

Robin Hirsch ,
Ian Hodkinson  and
Roger D. Maddux
Robin Hirsch
Affiliation:
Department of Computer Science, University College, Gower Street, London WC1E 6BT, UK, E-mail: R.Hirsch@cs.ucl.ac.uk
Ian Hodkinson
Affiliation:
Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK, E-mail: imh@doc.ic.ac.uk
Roger D. Maddux
Affiliation:
Department of Mathematics, 400 Carver Hall, Iowa State University, Ames, IOWA 50011-2066, USA, E-mail: maddux@iastate.edu
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Abstract

We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language ℒ with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 197 - 213
Copyright
Copyright © Association for Symbolic Logic 2002

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References

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