On $\boldsymbol{\Sigma}^1_3$- and $\Sigma^1_4$-uniformization

Abstract

Assuming the consistency of $\mathsf{ZFC}$, we construct a model of set theory in which the boldface $\mathbf{\Sigma}^1_3$-uniformization property holds, yet the lightface $\Sigma^1_4$-uniformization property fails, separating these two principles for the first time. We also indicate how to create a universe where $\Sigma^1_3$-uniformization holds, but $\Sigma^1_4$-uniformization fails using inner models with large cardinals.