Forcing the $\Pi^1_n$-Uniformization Property

TL;DR

This paper constructs models where the $oldsymbol{ m oldsymbol{ ext{Pi}}^1_n}$-uniformization property holds for various n, reducing the needed consistency strength and challenging existing patterns, using advanced forcing techniques.

Contribution

It introduces a forcing method to establish the $oldsymbol{ m oldsymbol{ ext{Pi}}^1_n}$-uniformization property in models with lower consistency strength and extends results to inner models with Woodin cardinals.

Findings

Established $oldsymbol{ m oldsymbol{ ext{Pi}}^1_3}$-uniformization in ZFC

Extended uniformization results to models with Woodin cardinals for $n > 1$

Produced models contradicting the natural $oldsymbol{ m oldsymbol{ ext{PD}}}$-induced pattern

Abstract

We generically construct a model in which the ${\Pi^1_3}$-uniformization property is true, thus lowering the best known consistency strength from the existence of $M_1^{\#}$ to just $\mathsf{ZFC}$. The forcing construction can be adapted to work over canonical inner models with Woodin cardinals, which yields, for the first time, universes where the $\Pi^1_{2n}$-uniformization property holds for $n >1$, thus producing models which contradict the natural $\mathsf{PD}$-induced pattern. It can also be used to obtain models for the $\Pi^1_1$-uniformization property in the generalized Baire space.