Forcing upper $Σ$-uniformization in the presence of lower $Π$-reduction or uniformization
Stefan Hoffelner
TL;DR
This paper develops a novel forcing technique that blends $\Pi^1_3$-reduction methods with $\Sigma^1_n$-uniformization to investigate the balance between reduction and uniformization in the projective hierarchy. By constructing intricate, layered forcings built from independent Suslin trees and almost-disjoint codings, the authors obtain models where $\Pi^1_3$-reduction holds while $\Pi^1_3$-uniformization fails and $\Sigma^1_n$-uniformization holds for all $n\ge 4$, and another model where $\Pi^1_3$-uniformization holds along with $\Sigma^1_n$-uniformization for all $n\ge 4$, with liftability to inner models $M_n$ containing Woodin cardinals. The work introduces a thinning-out (infinity-allows) framework to resolve fixed-point issues in coding forcings, thereby enabling controlled forcing that simultaneously handles both sides of the projective spectrum. In addition, the paper builds toward a $\Sigma^1_4$-uniformization result and a good $\Sigma^1_5$-well-order, broadening the landscape of possible global regularity phenomena without large-cardinal assumptions. The findings advance the understanding of how forcing can orchestrate complementary uniformization and reduction properties, with potential implications for further hierarchical refinements and inner-model liftings.
Abstract
We present a method which allows the combination of forcing uniformization on the $Π$- and the $Σ$-side of the projective hierarchy to a certain extent. Using this method we construct a universe where $Π^1_3$-reduction holds, $Π^1_3$-uniformization fails, yet $Σ^1_n$ uniformization is true for $n \ge 4$. We also construct a universe where $Π^1_3$-uniformization holds and for every $n \ge 4, $ $ Σ^1_4$-uniformization holds, lowering best known upper bound for this statement from the existence of two Woodin cardinals to $Con(\ZFC)$.