More on the preservation of large cardinals under class forcing

Journal of Symbolic Logic:1-34 (forthcoming)
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We prove two general results about the preservation of extendible and $C^{}$ -extendible cardinals under a wide class of forcing iterations. As applications we give new proofs of the preservation of Vopěnka’s Principle and $C^{}$ -extendible cardinals under Jensen’s iteration for forcing the GCH [17], previously obtained in [8, 27], respectively. We prove that $C^{}$ -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 $C^{}$ -extendible cardinals. We also show, assuming the GCH, that the class forcing iteration of Cummings–Foreman–Magidor for forcing $\diamondsuit _{\kappa ^+}^+$ at every $\kappa $ [10] preserves $C^{}$ -extendible cardinals. We give an optimal result on the consistency of weak square principles and $C^{}$ -extendible cardinals. In the last section prove another preservation result for $C^{}$ -extendible cardinals under very general class forcing iterations. As applications we prove the consistency of $C^{}$ -extendible cardinals with $\mathrm {{V}}=\mathrm {{HOD}}$, and also with $\mathrm {GA}$ plus $\mathrm {V}\neq \mathrm {HOD}$, the latter being a strengthening of a result from [14].



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The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
Squares, Scales and Stationary Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (01):35-98.
The Lottery Preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
Suitable Extender Models I.W. Hugh Woodin - 2010 - Journal of Mathematical Logic 10 (1):101-339.

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