More on the preservation of large cardinals under class forcing

Abstract

We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vop\v{e}nka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Delta_2$-definable behaviour of the power-set function on regular cardinals. We show that one can force proper class-many disagreements between the universe and HOD with respect to the calculation of successors of regular cardinals, while preserving $C^{(n)}$-extendible cardinals. We also show, assuming the GCH, that the class forcing iteration of Cummings-Foreman-Magidor for forcing $\diamondsuit_{\kappa^+}^+$ at every $\kappa$ preserves $C^{(n)}$-extendible cardinals. We give an optimal result on the consistency of weak square principles and $C^{(n)}$-extendible cardinals. In the last section we prove another preservation result for $C^{(n)}$-extendible cardinals under very general (not necessarily definable or weakly homogeneous) class forcing iterations. As applications we prove the consistency of $C^{(n)}$-extendible cardinals with $\rm{V}=\rm{HOD}$, and also with $\mathrm{GA}$ (the Ground Axiom) plus $\mathrm{V}\neq \mathrm{HOD}$, the latter being a strengthening of a result by Hamkins, Reitz and Woodin.