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More on the preservation of large cardinals under class forcing. (arXiv:1810.09195v1 [math.LO])
来源于:arXiv
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 查看全文>>