The Variety of Projection of a Tree-Prikry Forcing

Abstract

We study which κ-distributive forcing notions of size κ can be embedded into tree Prikry forcing notions with κ-complete ultrafilters under various large cardinal assumptions. An alternative formulation -can the filter of dense open subsets of a κ-distributive forcing notion of size κ be extended to a κ-complete ultrafilter. Definition 2. Let πWith the separative order p ≤ P/H q if an only if for every q ≤ P r, r is compatible with p.Claim 3. Let P, Q be any forcing notions, then:(1) Let G ⊆ P be V -generic and π :

On -Strongly Measurable Cardinals

“…Usually, in order to achieve this, some strong compactness assumption is made that enables one to embed any sufficiently distributive forcing into a Prikry-type forcing. See [14,6] for some examples for the consistency strength of such constructions.…”

Non-Galvin Filters

“…U is singular in M U , it follows that [id] U = κ δ for some limit δ since by [BGH21,Lemma 46] each successor element of the sequence of the form κ i+1 is regular in M U .…”

“…To deduce that o K (κ) ≥ 2, suppose otherwise that o K (κ) = 1, and denote by W the only measure on κ in K. Since M U is closed under ωsequences κ n | n < ω ∈ M U . Now all the κ n 's are critical points of the iteration, namely for some α n , crit(i αn,θ ) = κ n = i 0,αn (κ) (see for example [BGH21,Corollary 43]) and i αn,α n+1 is the ulrtapower embedding by i 0,αn (W ). Let α ω = sup n<ω α n .…”

On Cohen and Prikry Forcing Notions

“…Actually, more is true, under the assumption that κ is κ-compact there is a single Prikry-type forcing which absorbs all the κ-distributive forcings of cardinality κ (see [17]). In the absence of very large cardinals the situation changes, indeed, [9] Hayut and the authors proved that if a certain < κ-strategically closed forcing of cardinality κ is a projection of the tree-Prikry forcing then it is consistent that there is a cardinal λ with high Mitchell order, namely o(λ) > λ + . In [7], the authors proved that starting from a measurable cardinal (which is the minimal large cardinal assumption in the context of Prikry forcing) it is consistent that there is an (non-normal) ultrafilter U , such that the Prikry forcing with U projects onto the Cohen forcing Cohen(κ, 1), this was improved later in [9] to a larger class of forcing notions called Masterable forcings.…”

“…In the absence of very large cardinals the situation changes, indeed, [9] Hayut and the authors proved that if a certain < κ-strategically closed forcing of cardinality κ is a projection of the tree-Prikry forcing then it is consistent that there is a cardinal λ with high Mitchell order, namely o(λ) > λ + . In [7], the authors proved that starting from a measurable cardinal (which is the minimal large cardinal assumption in the context of Prikry forcing) it is consistent that there is an (non-normal) ultrafilter U , such that the Prikry forcing with U projects onto the Cohen forcing Cohen(κ, 1), this was improved later in [9] to a larger class of forcing notions called Masterable forcings. In the context of Prikry-type forcings, existence of such embeddings and projections allows one to iterate distributive forcing notions on different cardinals, see [15, Section 6.4].…”