Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Absoluteness via Resurrection

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.Comment: 34 page