Monotone versus non‐monotone projective operators

Abstract

For a class of operators , let denote the closure ordinal of ‐inductive definitions. We give upper bounds on the values of and under the assumption that all projective sets of reals are determined, significantly improving the known results. In particular, the bounds show that and hold for under the assumption of projective determinacy. Some of these inequalities were obtained by Aanderaa in the 70s via recursion‐theoretic methods but never appeared in print. Our proofs are model‐theoretic.