The Omega Rule is II_2^0-Hard in the lambda beta -Calculus

Abstract

We give a many-one reduction of the set of true /spl Pi//sub 2//sup 0/ sentences to the set of consequences of the lambda calculus with the omega rule. This solves in the affirmative a well known problem of H. Barendregt. The technique of proof has interest in itself and can be extended to prove that the theory which identifies all unsolvable terms together with the omega rule is H/sub 1//sup 1/-complete which solves another long-standing conjecture of H. Barendregt.