Collapse forcings are terminal for projective generic absoluteness: they yield the same invariance as all forcings under large cardinals, and the result holds in ZFC at low levels.
On forcing projective generic absoluteness from strong cardinals
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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 $\kappa_n$.
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2025 1verdicts
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Terminal Absoluteness of Collapse Forcings
Collapse forcings are terminal for projective generic absoluteness: they yield the same invariance as all forcings under large cardinals, and the result holds in ZFC at low levels.