Indestructible Strong Unfoldability

Abstract

Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all <$\kappa$-closed ${\kappa }^{+}$-preserving forcing. This degree of indestructibility, we prove, is the best possible from this hypothesis within the class of <$\kappa$-closed forcing. From a stronger hypothesis, however, we prove that the strong unfoldability of $\kappa$ can be made indestructible by all <$\kappa$-closed forcing. Such indestructibility, we prove, does not follow from indestructibility merely by <$\kappa$-directed closed forcing. Finally, we obtain global and universal forms of indestructibility for strong unfoldability, finding the exact consistency strength of universal indestructibility for strong unfoldability.