Forcing the $\Pi^1_3$-Reduction Property and a Failure of $\Pi^1_3$-Uniformization

Abstract

We force over the constructible universe to obtain a model of the $\Pi^1_3$-reduction property, thus lowering the best known large cardinal strength from the existence of $M_1^{\#}$ to just ZFC. In this model the $\Pi^1_3$-uniformization property fails, which separates these two principles for the first time.