More on the preservation of large cardinals under class forcing

TL;DR

This paper establishes broad preservation results for large cardinals, specifically extendible and $C^{(n)}$-extendible cardinals, under various class forcing iterations, with applications to principles like Vopenka's and the Ground Axiom.

Contribution

It provides new general preservation theorems for large cardinals under class forcing, extending previous results and applying to diverse forcing scenarios including Easton-support and class iterations.

Findings

Preservation of $C^{(n)}$-extendible cardinals under Easton-support iterations.

Forcing with class iterations can produce many disagreements between universe and HOD.

Consistency of $C^{(n)}$-extendible cardinals with V=HOD and the Ground Axiom.

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.