# Relation algebras from cylindric algebras, II

Robin Hirsch and Ian Hodkinson

We prove, for each 4 \leq n <\omega, that **S Ra CA _{n+1 }** cannot be defined, using only finitely many first-order axioms, relative to

**S Ra CA**. The construction also shows that for 3 \leq m<n<\omega,

_{n}**S Nr**is not finitely axiomatisable over

_{m}CA_{n+1}**S Nr**. In consequence, for a certain standard n-variable first-order proof system |-m,n of m-variable formulas, there is no finite set of m-variable schemata whose m-variable instances, when added to |-

_{m}CA_{n}_{m,n}as axioms, yield |-

_{m,n+1}.