Relation algebras form cylindric algebras, II

We prove, for each 4 ≤ n < ω, that SRaCAn+1 cannot be defined, using only finitely many first-order axioms, relative to SRaCAn. The construction also shows that for 5 ≤ n < ω, SRaCAn is not finitely axiomatisable over RAn, and that for 3 ≤ m < n < ω, SNrmCAn+1 is not finitely axiomatisable over SNrmCAn. 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,n as axioms, yield `m,n+1. AMS classification 03G15; 03C05, 03C25, 03F20.