Splitting stationary sets in

2012 ◽  
Vol 77 (1) ◽  
pp. 49-62 ◽  
Author(s):  
Toshimichi Usuba
Keyword(s):  
Uncountable Cardinal ◽  
Stationary Set ◽  
Stationary Subsets

AbstractLet A be a non-empty set. A set is said to be stationary in if for every f: [A]<ω → A there exists x ϵ S such that x ≠ A and f“[x]<ω ⊆ x. In this paper we prove the following: For an uncountable cardinal λ and a stationary set S in , if there is a regular uncountable cardinal κ ≤ λ such that {x ϵ S: x ∩ κ ϵ κ} is stationary, then S can be split into κ disjoint stationary subsets.

On splitting stationary subsets of large cardinals

1977 ◽  
Vol 42 (2) ◽  
pp. 203-214 ◽  
Author(s):  
James E. Baumgartner ◽  
Alan D. Taylor ◽  
Stanley Wagon
Keyword(s):  
Large Cardinals ◽  
Normal Ideal ◽  
Weakly Compact ◽  
Uncountable Cardinal ◽  
Stationary Set ◽  
Stationary Subsets ◽  
Open Question ◽  
Mahlo Cardinals ◽  
Normal Ideals

AbstractLet κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ+-saturated, i.e., are there κ+ stationary subsets of κ with pairwise intersections nonstationary? Our first observation is:Theorem. NS isκ+-saturated iff for every normal ideal J on κ there is a stationary set A ⊆ κsuch that J = NS∣A = {X ⊆ κ: X ∩ A ∈ NS}.Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define “greatly Mahlo” cardinals and obtain the following:Theorem. If κ is greatly Mahlo then NS is notκ+-saturated.Theorem. If κ is ordinal Π11-indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries aκ-saturated ideal, thenκis greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π11-indescribable cardinal.These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ+-saturated.

scholarly journals Possible behaviours of the reflection ordering of stationary sets

1995 ◽  
Vol 60 (2) ◽  
pp. 534-547 ◽  
Author(s):  
Jiří Witzany
Keyword(s):  
Partial Ordering ◽  
Generic Extension ◽  
Maximal Antichain ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Stationary Subsets ◽  
Almost All

AbstractIf S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so thatWe prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

scholarly journals Sigma-Prikry forcing II: Iteration Scheme

2021 ◽  
pp. 2150019
Author(s):  
Alejandro Poveda ◽  
Assaf Rinot ◽  
Dima Sinapova
Keyword(s):  
Blow Up ◽  
General Scheme ◽  
Iteration Scheme ◽  
Singular Cardinal ◽  
Prikry Forcing ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis ◽  
Singular Cardinals ◽  
Stationary Set ◽  
Stationary Subsets

In Part I of this series [5], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets, as well as verify that the Extender-based Prikry forcing is [Formula: see text]-Prikry. As an application, we blow-up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all nonreflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.

Ideals and combinatorial principles

1997 ◽  
Vol 62 (1) ◽  
pp. 117-122 ◽  
Author(s):  
Douglas Burke ◽  
Yo Matsubara
Keyword(s):  
Regular Cardinal ◽  
Stationary Subset ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Uncountable Cardinal ◽  
Combinatorial Principles ◽  
Stationary Subsets

It is well known that if σ is a strongly compact cardinal and λ a regular cardinal ≥ σ, then for every stationary subset X of {α < λ: cof (α) = ω} there is some β < λ such that X ⋂ β is stationary in β. In fact the existence of a uniform, countably complete ultrafilter over λ is sufficient to prove the same conclusion about stationary subsets of {α < λ: cof (α) = ω}. See [13] or [10]. By analyzing the proof of this theorem as presented in [10], we realized the same conclusion will follow from the existence of a certain ideal, not necessarily prime, on . Throughout we will assume that σ is a regular uncountable cardinal and use the word “ideal” to mean fine ideal.

Strong analogues of Martin's axiom Imply Axiom R

1987 ◽  
Vol 52 (1) ◽  
pp. 216-218 ◽  
Author(s):  
Robert E. Beaudoin
Keyword(s):  
Reflection Principle ◽  
Partial Orders ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Stationary Set ◽  
Stationary Subsets

AbstractWe show that either PFA+ or Martin's maximum implies Fleissner's Axiom R, a reflection principle for stationary subsets of Pℵ(λ). In fact, the “plus version” (for one term denoting a stationary set) of Martin's axiom for countably closed partial orders implies Axiom R.

scholarly journals Full reflection at a measurable cardinal

1994 ◽  
Vol 59 (2) ◽  
pp. 615-630 ◽  
Author(s):  
Thomas Jech ◽  
Jiří Witzany
Keyword(s):  
Measurable Cardinal ◽  
Higher Order ◽  
Stationary Subset ◽  
Regular Cardinals ◽  
Uncountable Cardinal ◽  
Stationary Set ◽  
Full Reflection

AbstractA stationary subset S of a regular uncountable cardinal κreflects fully at regular cardinals if for every stationary set T ⊆ κ of higher order consisting of regular cardinals there exists an α Є T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection holds at every λ ≤ κ and κ is measurable.

scholarly journals Sigma-Prikry forcing I: The Axioms

2020 ◽  
pp. 1-34
Author(s):  
Alejandro Poveda ◽  
Assaf Rinot ◽  
Dima Sinapova
Keyword(s):  
Iteration Scheme ◽  
Finite Collection ◽  
Strong Limit ◽  
Prikry Forcing ◽  
Supercompact Cardinals ◽  
Limit Cardinal ◽  
Singular Cardinals ◽  
Stationary Set ◽  
Stationary Subsets

Abstract We introduce a class of notions of forcing which we call $\Sigma $ -Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma $ -Prikry. We show that given a $\Sigma $ -Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set T, there exists a corresponding $\Sigma $ -Prikry poset that projects to $\mathbb P$ and kills the stationarity of T. Then, in a sequel to this paper, we develop an iteration scheme for $\Sigma $ -Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $\kappa $ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which $\kappa $ remains a strong limit cardinal, every finite collection of stationary subsets of $\kappa ^+$ reflects simultaneously, and $2^\kappa =\kappa ^{++}$ .

On a topological construction of Juhasz and Shelah

1992 ◽  
Vol 57 (1) ◽  
pp. 166-171
Author(s):  
Dan Velleman
Keyword(s):  
Topological Space ◽  
Additional Structure ◽  
Homogeneous Set ◽  
Stationary Set ◽  
Topological Construction

In [2], Juhasz and Shelah use a forcing argument to show that it is consistent with GCH that there is a 0-dimensional T2 topological space X of cardinality ℵ3 such that every partition of the triples of X into countably many pieces has a nondiscrete (in the topology) homogeneous set. In this paper we will show how to construct such a space using a simplified (ω2, 1)-morass with certain additional structure added to it. The additional structure will be a slight strengthening of a built-in ◊ sequence, analogous to the strengthening of ordinary ◊k to ◊S for a stationary set S ⊆ k.Suppose 〈〈θα∣ ∝ ≤ ω2〉, 〈∝β∣α < β ≤ ω2〉〉 is a neat simplified (ω2, 1)-morass (see [3]). Let ℒ be a language with countably many symbols of all types, and suppose that for each α < ω2, α is an ℒ-structure with universe θα. The sequence 〈α∣α < ω2 is called a built-in ◊ sequence for the morass if for every ℒ-structure with universe ω3 there is some α < ω2 and some f ∈αω2 such that f(α) ≺ , where f(α) is the ℒ-structure isomorphic to α under the isomorphism f. We can strengthen this slightly by assuming that α is only defined for α ∈ S, for some stationary set S ⊆ ω2. We will then say that is a built-in ◊ sequence on levels in S if for every ℒ-structure with universe ω3 there is some α ∈ S and some f ∈ αω2 such that f(α) ≺ .

Co-stationarity of the ground model

2006 ◽  
Vol 71 (3) ◽  
pp. 1029-1043 ◽  
Author(s):  
Natasha Dobrinen ◽  
Sy-David Friedman
Keyword(s):  
Large Cardinals ◽  
Partial Ordering ◽  
Proper Class ◽  
Ground Model ◽  
Covering Theorem ◽  
Uncountable Cardinal ◽  
Cohen Forcing ◽  
Measurable Cardinals

AbstractThis paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

Increasing u2 by a stationary set preserving forcing

2009 ◽  
Vol 74 (1) ◽  
pp. 187-200
Author(s):  
Benjamin Claverie ◽  
Ralf Schindler
Keyword(s):  
Regular Cardinal ◽  
Nonstationary Ideal ◽  
Stationary Set ◽  
Precipitous Ideal ◽  
Countable Structure

AbstractWe show that if I is a precipitous ideal on ω1 and if θ > ω1 is a regular cardinal, then there is a forcing ℙ = ℙ(I, θ) which preserves the stationarity of all I-positive sets such that in Vℙ, ⟨Hθ; ∈, I⟩ is a generic iterate of a countable structure ⟨M; ∈, Ī⟩. This shows that if the nonstationary ideal on ω1 is precipitous and exists, then there is a stationary set preserving forcing which increases . Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω1 is precipitous, then .

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