A Universe with a $\Delta^1_n$-definable well-order of the reals, $\mathsf{CH}$ and $\Pi^1_n$-Uniformization

Abstract

This paper details the construction of a universe where $\Pi^1_3$-uniformization is true, the Continuum Hypothesis holds yet it possesses a $\Delta^1_3$-definable well-order of its reals. The method can be lifted to canonical inner models with finitely many Woodin cardinals to produce universes of $\mathsf{CH}$, $\Pi^1_n$-uniformization and where additionally a $\Delta^1_n$-definable well-order of the reals exist.