More on the preservation of large cardinals under class forcing

Bagaria Joan, Poveda Alejandro

Published 2018-10-22Version 1

We introduce first the large-cardinal notion of $\Sigma_n$-supercompactness as a higher-level analog of the well-known Magidor's characterization of supercompact cardinals, and show that a cardinal is $C^{(n)}$-extendible if and only if it is $\Sigma_{n+1}$-supercompact. This yields a new characterization of $C^{(n)}$-extendible cardinals which underlines their role as natural milestones in the region of the large-cardinal hierarchy between the first supercompact cardinal and Vop\v{e}nka's Principle ($\rm{VP}$). We then develop a general setting for the preservation of $\Sigma_n$-supercompact cardinals under class forcing iterations. As a result we obtain new proofs of the consistency of the GCH with $C^{(n)}$-extendible cardinals (cf.~\cite{Tsa13}) and the consistency of $\rm{VP}$ with the GCH (cf.~\cite{Broo}). Further, we show that $C^{(n)}$-extendible cardinals are preserved after forcing with standard Easton class forcing iterations for any $\Pi_1$-definable possible behaviour of the power-set function on regular cardinals, and that $\rm{VP}$ is preserved by any definable such iteration. Finally, we show that, assuming the GCH, the class forcing iteration of Cummings-Foreman-Magidor forcing notions for forcing $\diamondsuit_{\kappa^+}^+$ at every $\kappa$ (\cite{CummingsSquare}) preserves $C^{(n)}$-extendible cardinals, hence it also preserves $\rm{VP}$.