Indestructible Strong Unfoldability

Using the lottery preparation, we prove that any strongly unfoldable cardinal κ can be made indestructible by all <κ-closed κ+-preserving forcing. This degree of indestructibility, we prove, is the best possible from this hypothesis within the class of <κ-closed forcing. From a stronger hypothesis, however, we prove that the strong unfoldability of κ can be made indestructible by all <κ-closed forcing. Such indestructibility, we prove, does not follow from indestructibility merely by <κ-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. The unfoldable cardinals were introduced by Villaveces in [Vil98], along with their companion notion, the strongly unfoldable cardinals, which turn out to be the same as what Miyamoto [Miy98] independently introduced as the Hκ+ reflecting cardinals. These cardinals lie relatively low in the large cardinal hierarchy, somewhat above the weakly compact cardinals, and they relativize to L in the sense that every unfoldable cardinal is unfoldable in L and in fact strongly unfoldable there, as in L the two notions coincide. For this reason, the notions of unfoldability and strong unfoldability, although not equivalent, have the same consistency strength, bounded below by the totally indescribable cardinals and above by the