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

Abstract

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