Monotone versus non‐monotone projective operators

For a class of operators Γ $\Gamma$ , let | Γ | $|\Gamma |$ denote the closure ordinal of Γ $\Gamma$ ‐inductive definitions. We give upper bounds on the values of | Σ 2 n + 1 1 , m o n | $|\Sigma ^{1,mon}_{2n+1}|$ and | Π 2 n + 2 1 , m o n | $|\Pi ^{1,mon}_{2n+2}|$ under the assumption that all projective sets of reals are determined, significantly improving the known results. In particular, the bounds show that | Π n 1 , m o n | < | Π n 1 | $|\Pi ^{1,mon}_{n}| < |\Pi ^1_{n}|$ and | Σ n 1 , m o n | < | Σ n 1 | $|\Sigma ^{1,mon}_{n}| < |\Sigma ^1_{n}|$ hold for 2 ⩽ n $2\leqslant n$ 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.