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

Abstract

We show that one can force over $L$ that $\Sigma^1_3$-separation holds, while $\Pi^1_3$-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 $M_n$, $\Sigma^1_{n+3}$-separation can hold, yet $\Pi^1_{n+3}$-reduction fails.