  W.H. Woodin showed that if $\kappa_1 < \cdots < \kappa_n$ are strong
cardinals then two-step ${\bf\Sigma}^1_{n+3}$ generic absoluteness holds after
collapsing $2^{2^{\kappa_n}}$ to be countable. We show that this number can be
reduced to $2^{\kappa_n}$, and to $\kappa_n^+$ in the case $n = 1$, but cannot
be further reduced to $\kappa_n$.

Logic