Strongly unfoldable cardinals made indestructible

Abstract

I provide indestructibility results for large cardinals consistent with V=L, such as weakly compact, indescribable and strongly unfoldable cardinals. The Main Theorem shows that any strongly unfoldable cardinal κ can be made indestructible by < κ-closed, κ-proper forcing. This class of posets includes for instance all < κ-closed posets that are either κ⁺-c.c. or ≤ κ-strategically closed as well as finite iterations of such posets. Since strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, the Main Theorem therefore makes these two large cardinal notions similarly indestructible. Finally, I apply the Main Theorem to obtain a class forcing extension preserving all strongly unfoldable cardinals in which every strongly unfoldable cardinal κ is indestructible by < κ-closed, κ-proper forcing.