On forcing projective generic absoluteness from strong cardinals

TL;DR

This paper improves the known large cardinal assumptions needed for certain generic absoluteness results, reducing the collapse size from a double exponential to a single exponential or successor cardinal, with specific cases analyzed.

Contribution

It demonstrates that the collapse size for obtaining projective generic absoluteness from strong cardinals can be significantly reduced, refining previous results.

Findings

Collapse size reduced from double exponential to exponential in strong cardinals

Achieves collapse to $\kappa_n^+$ for the case n=1

Shows further reduction to $\kappa_n$ is impossible

Abstract

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 $\kappa_n$.