Absoluteness via resurrection

摘要

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Velickovic. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms RA(alpha)(Gamma) for a class of forcings Gamma and a given ordinal alpha), and show that 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).