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

TL;DR

This paper constructs a universe of set theory where the Continuum Hypothesis holds, a specific uniformization principle is true, and there exists a well-order of the reals that is definable at a certain complexity level, advancing understanding of definability and uniformization.

Contribution

It introduces a method to build models with CH, $oldsymbol{ ext{Pi}}^1_n$-uniformization, and a $oldsymbol{ ext{Delta}}^1_n$-definable well-order of the reals, extending to inner models with Woodin cardinals.

Findings

Existence of a universe with CH and a $oldsymbol{ ext{Delta}}^1_3$-definable well-order of the reals.

Construction of models satisfying $oldsymbol{ ext{Pi}}^1_3$-uniformization.

Method extends to inner models with finitely many Woodin cardinals.

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.