Absoluteness via Resurrection

Abstract

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Velickovi\'c. We introduce a stronger form of resurrection axioms (the \emph{iterated} resurrection axioms $\textrm{RA}_\alpha(\Gamma)$ for a class of forcings $\Gamma$ and a given ordinal $\alpha$), and show that $\textrm{RA}_\omega(\Gamma)$ implies generic absoluteness for the first-order theory of $H_{\gamma^+}$ with respect to forcings in $\Gamma$ preserving the axiom, where $\gamma=\gamma_\Gamma$ is a cardinal which depends on $\Gamma$ ($\gamma_\Gamma=\omega_1$ if $\Gamma$ is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover we outline that simultaneous generic absoluteness for $H_{\gamma_0^+}$ with respect to $\Gamma_0$ and for $H_{\gamma_1^+}$ with respect to $\Gamma_1$ with $\gamma_0=\gamma_{\Gamma_0}\neq\gamma_{\Gamma_1}=\gamma_1$ is in principle possible, and we present several natural models of the Morse Kelley set theory where this phenomenon occurs (even for all $H_\gamma$ simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.