Indestructibility, strongness, and level by level equivalence

Abstract

Abstract. We construct a model in which there is a strong cardinal κ whose strongness is indestructible under κ-strategically closed forcing and in which level by level equivalence between strong compactness and supercompactness holds non-trivially. Introduction and preliminaries.A very surprising fact (see ) is that if there are large enough cardinals in the universe, then indestructibility for either a strong or supercompact cardinal (in the sense of [5] or [13]) is incompatible with level by level equivalence between strong compactness and supercompactness. Indeed, Theorem 6 of [3] states that if κ is a strong cardinal such that forcing with any κ-strategically closed partial ordering preserves κ's strongness (where for the rest of this paper, we will refer to such a cardinal as an indestructible strong cardinal ) and level by level equivalence between strong compactness and supercompactness holds below κ (where for the rest of this paper, level by level equivalence between strong compactness and supercompactness means that for δ ≤ λ, δ and λ both regular, δ is λ strongly compact iff δ is λ supercompact), then no cardinal λ > κ is 2 λ supercompact. However, a question left unanswered in [3] is whether it is possible for there to be level by level equivalence non-trivially below an indestructible strong cardinal κ in a universe in which no cardinal λ > κ is 2 λ supercompact. By the next to last sentence, the existence of such a universe is the best possible outcome for which one could hope.The purpose of this paper is to provide an affirmative answer to the aforementioned question. Specifically, we prove the following theorem.

Indestructibility, instances of strong compactness, and level by level inequivalence

“…We 3 Although not directly relevant to this proof, it is actually true that P "Ṗ is ℵ 2 -directed closed". 4 An outline of this argument is as follows. will not provide as many details as in [5], although we will give a reasonably complete proof.…”

“…For instance, in [8,Theorems 5 and 6], it was shown that if κ < λ are such that κ is either indestructibly supercompact or indestructibly strong and λ is 2 λ supercompact, then B = {δ < κ | δ is δ + strongly compact yet δ is not δ + supercompact} must be unbounded in κ. However, if the universe contains a fairly small number of large cardinals, κ is either strong or supercompact, and κ's supercompactness or strongness exhibits enough indestructibility, then [8,Theorem 8] and [4,Theorem 1] indicate that B may be empty. This sort of occurrence has been further examined in [1,2,3].…”

A Note on Indestructibility and Strong Compactness

“…To handle when this occurs, we use an idea due to Hamkins, which has also appeared in [5] in a more general context (as well as in this context in [1,Lemma 2.3]). Hamkins' argument is as follows.…”