A Failure of ${\Pi }_{n+3}^1$-Reduction in the Presence of ${\Sigma }_{n+3}^1$-Separation

Abstract

We show that one can force over $L$ that ${\Sigma }_3^1$-separation holds, while ${\Pi }_3^1$-reduction fails, thus separating these two principles for the first time. The construction can be lifted to canonical inner models $M_n$ with $n$-many Woodin cardinals, yielding that assuming the existence of $M_n$, ${\Sigma }_{n+3}^1$-separation can hold, yet ${\Pi }_{n+3}^1$-reduction fails.

arXiv comment41 pages