Monotone versus non-monotone projective operators

Abstract

For a class of operators $\Gamma$, let $\left|\Gamma \right|$ denote the closure ordinal of $\Gamma$-inductive definitions. We give upper bounds on the values of $|{\Sigma }_{2n+1}^{1,mon}|$ and $|{\Pi }_{2n+2}^{1,mon}|$ 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 $2⩽n$ 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.