4 problems

Let ${\mathbf{\delta}}^1_{2n+2}$ denote the supremum of the lengths of ${\mathbf{\Delta}}^1_{2n+2}$ pre-well-orderings of ${\mathbb{R}}$. Assume ${\sf{ZF+AD+DC}}$. Kechris's second…

Let ${\mathbf{\delta}}^1_{2n+2}$ denote the supremum of the lengths of ${\mathbf{\Delta}}^1_{2n+2}$ pre-well-orderings of ${\mathbb{R}}$. Assume ${\sf{ZF+AD+DC}}$. Kechris's first…

Let $\lambda$ be a cardinal with $\bf{\delta^1_4}\leq\lambda\leq\bf{\delta^1_6}$, and suppose $\operatorname{cof}(\lambda)>\omega$. A cardinal is $\bf{\Gamma}$-unreachable when the…

Let $\bf{\Gamma}$ be a pointclass, and let $\kappa$ be $\bf{\Gamma}$-reachable when there is a sequence of distinct $\bf{\Gamma}$-sets of length $\kappa$; otherwise, $\kappa$ is…