Forcing upper $\Sigma$-uniformization in the presence of lower $\Pi$-reduction or uniformization

Abstract

We present a method which allows the combination of forcing uniformization on the $\Pi$- and the $\Sigma$-side of the projective hierarchy to a certain extent. Using this method we construct a universe where ${\Pi }_3^1$-reduction holds, ${\Pi }_3^1$-uniformization fails, yet ${\Sigma }_n^1$ uniformization is true for $n\ge 4$. We also construct a universe where ${\Pi }_3^1$-uniformization holds and for every $n\ge 4,$ ${\Sigma }_4^1$-uniformization holds, lowering best known upper bound for this statement from the existence of two Woodin cardinals to $Con(\ZFC)$.

arXiv comment70 pages