On certain indestructibility of strong cardinals and a question of Hajnal

On tall cardinals and some related generalizations

“…As in [9], we will say that the partial ordering P is κ + -weakly closed and satisfies the Prikry property if it meets the following criteria.…”

“…To date, this question remains open, and has defied every effort to obtain a positive answer. We were able to prove Theorem 1 because the work of [9] shows that it is possible to do Prikry forcing above a strong cardinal while preserving strongness. However, as is fairly well known (see, e.g., [16,Section 4] and [3, Lemma 3.1]), adding a Prikry sequence above a strongly compact cardinal destroys strong compactness.…”

“…It will then be the case that (j(κ)) ω ⊆ M but (j(λ)) ω ⊆ M . In addition, an argument using the work of [9] shows that it is impossible to replace the condition of Theorem 4(3) with η α = j(f )(ξ 1 , . .…”

Indestructibility under adding Cohen subsets and level by level equivalence

“…Its definition is given in [13], [14], and [10], but for concreteness, we repeat it here. Specifically, let p, q ∈ P, p = ṗ α : α < κ , q = q α : α < κ .…”

Indestructibility, strongness, and level by level equivalence

“…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.…”