Squares and uncountably singularized cardinals

Abstract

It is known that if κ is inaccessible in V , and W is an outer model of V such that (κ +) V = (κ +) W , and cf W (κ) = ω, then κ,ω holds in W. Many strengthenings of this theorem have been investigated as well. We show that this theorem does not generalize to uncountable cofinalities: There is a model V in which κ is inaccessible and there is a forcing extension W of V in which (κ +) V = (κ +) W , ω < cf W (κ) < κ, and κ,τ fails in W for all τ < κ. We make use of Magidor's forcing for singularizing an inaccessible κ to have uncountable cofinality. Along the way, we analyze stationary reflection in this model, and we show that it is possible for κ,cf(κ) to hold in a forcing extension by Magidor's poset if the ground model is prepared with a partial square sequence.

Knaster and friends III: Subadditive colorings

“…We shall now turn to prove Clause (3) of Theorem C, in particular, establishing that, in general, for uncountable θ, ⊟ ind (κ, θ) is not equivalent to ind (κ, θ). This will follow from the following two theorems; these are fairly straightforward modifications of results of Cummings and Schimmerling [CS02] and Levine and Sinapova [LS21], respectively, but we provide some details for completeness.…”

“…Therefore, in V C * Ṁ, any element of D δ is a thread through D * . However, the proof of [LS21,Lemma 4.7] shows that forcing over V C δ * Ṁδ with (C * Ṁ)/(C δ * Ṁδ ) cannot add a thread to a (δ, τ )-sequence. ([LS21, Lemma 4.7] is about λ,τ -sequences, but the exact same proof still works for (δ, τ )-sequences.)…”

Knaster and Friends Iii: Subadditive Colorings

“…By forcing with the Laver preparation forcing if necessary, we may assume that the supercompactness of is indestructible under -directed closed forcing. Following [23, Section 4], let , and let be a -name for the Magidor forcing that turns into a singular cardinal of cofinality .…”

“…For every inaccessible above , let . By [23, Proposition 4.3], there is a club such that, for every inaccessible , is a -name for a Magidor forcing to turn into a singular cardinal of cofinality such that there is a complete embedding of into .…”

“…Therefore, in , any element of is a thread through . However, the proof of [23, Lemma 4.7] shows that forcing over with cannot add a thread to a -sequence. ([23, Lemma 4.7] is about -sequences, but the exact same proof still works for -sequences.)…”