Abstract
We prove, for each 4⩽ n< ω, that S Ra CA n+1 cannot be defined, using only finitely many first-order axioms, relative to S Ra CA n . The construction also shows that for 5⩽n<ω, S Ra CA n is not finitely axiomatisable over RA n , and that for 3⩽m<n<ω, S Nr m CA n+1 is not finitely axiomatisable over S Nr m CA n . 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 .
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