Solution of a Problem of Barendregt on Sensible lambda-Theories

H is the theory extending β-conversion by identifying all closed unsolvables. H ω is the closure of this theory under the ω-rule (and β-conversion). A long-standing conjecture of H. Barendregt states that the provable equations of H ω form Π 1 1 -complete set. Here we prove that conjecture.