There is a balance between the amount of (weak) indestructibility one can have and the amount of strong cardinals. It’s consistent relative to large cardinals to have lots of strong cardinals and all of their degrees of strength are weakly indestructible. But this necessitates the destructibility of the partially strong cardinals. Guaranteeing the indestructibility of the partially strong cardinals is shown to be harder. In particular, this work establishes an equiconsistency between: 1.a proper class of cardinals that are strong reflecting (...) strongs; and2.weak indestructibility for (κ+2)-strength for all cardinals κ in the presence of a proper class of strong cardinals.These have a much higher consistency strength than: 3.weak indestructibility for all degrees of strength for a proper class of strong cardinals.This discrepancy holds even if we weaken (2) from the presence of a proper class to just two strong cardinals. (2) is also equivalent to weak indestructibility for all λ-strength for λ far beyond (κ+2); well beyond the next measurable limit of measurables above κ, but before the next μ that is (μ+2)-strong.One direction of the equiconsistency of (1) and (2) is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed, and similar results are derived for supercompacts and supercompacts reflecting supercompacts.Abstract prepared by James Holland.E-mail: [email protected]: https://sites.math.rutgers.edu/~jch258/assets/dis.pdf. (shrink)

Marcel Proust, as a writer, was even more, shall we say , “steeped” in literary work by others than were so steeped such equally “classic,” as they are by now called, or “canonical” twentieth-century novelists as James Joyce, who, like Proust, seems to have had but in fact did not have a photographic memory, and Vladimir Nabokov, who did have one, or than are such literary critics as ..