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

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Contribution

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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 ${\Pi}^1_3$-reduction holds, $\Pi^1_3$-uniformization fails, yet $\Sigma^1_n$ uniformization is true for $n \ge 4$. We also construct a universe where $\Pi^1_3$-uniformization holds and for every $n \ge 4, $ $ \Sigma^1_4$-uniformization holds, lowering best known upper bound for this statement from the existence of two Woodin cardinals to $Con(\ZFC)$.