Forcing closed unbounded sets

Abraham, Uri; Shelah, Saharon

doi:10.2307/2273456

Cited by 71 publications

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References 10 publications

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“…Let us refer to the members of M κ X as κ-meagre subsets of X . This way, the classical meagre ideal on X is M ω 1 X . I will also denote it by M X .…”

Section: Fa(b) λ and The Covering Number Of The Meagre Ideal Of C ωmentioning

confidence: 99%

“…There is a particularly simple forcing notion for adding a club subset of ω 1 . This forcing was first defined and studied by Baumgartner [4] and will be denoted in this note by B.…”

Section: Introductionmentioning

confidence: 99%

“…This forcing was first defined and studied by Baumgartner [4] and will be denoted in this note by B. B is the set, ordered by reverse inclusion, of all finite functions f ⊆ ω 1 ×ω 1 that can be extended to a strictly increasing and continuous function F : ω 1 −→ ω 1 . B has size ℵ 1 , and the union of any generic filter for B is the enumerating function of a new club of ω 1 , which I will call a Baumgartner club (over V).…”

Section: Introductionmentioning

confidence: 99%

“…B is often described as the forcing for adding a club of ω 1 with finite conditions. It can be presented in other appealing ways too (see for example [1] or [16]). B is proper and so, since it has size ℵ 1 , preserves all cardinals; in fact, given any countable N H (ω 2 ) and any p ∈ B ∩ N , p * := p ∪ { δ N , δ N } is an (N , B)-generic condition extending p (see [4]) where, here and throughout the note, δ X denotes X ∩ ω 1 whenever X is a set and X ∩ ω 1 is an ordinal.…”

Section: Introductionmentioning

confidence: 99%

“…To point out some obvious examples, both forcing notions add the relevant new object by finite approximations, both are homogeneous (in the above sense) and of the least possible size, both have a simple definition, 1 and in fact both are absolute in a strong sense (in the sense that any two transitive models of ZF have the same Cohen forcing, and have the same B in case they agree on ω 1 ). It is therefore natural to ask if there is version of Add(ω, X ) for B.…”

Section: Introductionmentioning

confidence: 99%

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Adding many Baumgartner clubs

Asperó

2017

Arch. Math. Logic

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“…Let us refer to the members of M κ X as κ-meagre subsets of X . This way, the classical meagre ideal on X is M ω 1 X . I will also denote it by M X .…”

Section: Fa(b) λ and The Covering Number Of The Meagre Ideal Of C ωmentioning

confidence: 99%

“…There is a particularly simple forcing notion for adding a club subset of ω 1 . This forcing was first defined and studied by Baumgartner [4] and will be denoted in this note by B.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Adding many Baumgartner clubs

Asperó

2017

Arch. Math. Logic

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Definability degrees

Friedman

2005

Mathematical Logic Qtrly

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Key words Descriptive set theory, large cardinal, inner model. MSC (2000) 03E35, 03E45, 03E55We establish the equiconsistency of a simple statement in definability theory with the failure of the GCH at all infinite cardinals. The latter was shown by Foreman and Woodin ([2]) to be consistent, relative to the existence of large cardinals. Definition Suppose that κ is an infinite cardinal and A, B are subsets of κ. We say thatWe consider the following statement:Theorem 1 ( * ) κ holds for regular κ.P r o o f. Let C be a subset of κ and let P be κ-Cohen forcing over the ground model L [C]. If (A, B) is P × P -generic over L [C], then it is easy to verify that (A, C) and (B, C) are κ-incomparable. As only genericity over L κ+1 [C] is required, there exist such A, B in V .Theorem 2 Suppose that κ is a singular strong limit cardinal of uncountable cofinality and ( * ) κ holds. Then GCH fails at CUB-manyκ < κ.

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A general Mitchell style iteration

Krueger

2008

Mathematical Logic Qtrly

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Abstract. We work out the details of a schema for a mixed support forcing iteration, which generalizes the Mitchell model [7] with no Aronszajn trees on ω 2 .The main purpose of this paper is to present the details of a schema for a mixed support forcing iteration. This schema will provide the technical framework for a variety of consistency results which we establish in [6] and [4].In [5] we constructed a model in which the properties of being "internally club" and "internally approachable" are distinct, for structures of size the successor of a regular cardinal. Our construction was reminiscent in some ways of Mitchell's classic model with no Aronszajn trees on ω 2 ([7]). We used a forcing iteration whose factors are two-step iterations of Cohen forcing followed by a kind of collapse forcing, where the support on the Cohen part and on the collapsing part are of different sizes. However the kind of collapse forcing we used is not even strategically closed.In Section 2 we describe an iterable property which is satisfied by both the forcing used to construct Mitchell's model and the forcing we used in [5]. It is this property which we will assume to hold for the factor forcings in our mixed support iteration schema. We work out the details of this iteration in Section 3.In subsequent papers we use the iterated forcing schema presented below as a tool for establishing consistency results. In [6] we construct a model with a disjoint stationary sequence on the successor of an arbitrary regular cardinal, and also distinguish several variations of the property of being internally approachable for sets of a variety of sizes. In [4] we use the forcing iteration schema to study the approachability ideal at the second successor of a singular cardinal. PreliminariesWe review some background material, notation, and conventions used in the paper. We assume the reader has had some previous exposure to proper forcing and iterated forcing. We refer the reader to [2] for basic facts concerning iterated forcing. In the rest of the section, µ and κ always refer to regular cardinals.A forcing poset is a pair P, ≤ such that ≤ is a binary relation on P which is reflexive and transitive. Due to our treatment of iterated forcing, we do not require anti-symmetry as part of the definition of a forcing poset. In the case that p ≤ q and q ≤ p, we say that p and q are equivalent.A forcing poset P is µ-closed if whenever p i : i < ξ is a descending sequence of conditions in P for some ξ < µ, then there is q in P such that q ≤ p i for all i < ξ. The poset P satisfies the κ-covering property if it forces that whenever x is a set of ordinals in the extension with size less than κ, then there is a set a in the ground model with size less than κ in the ground model such that x ⊆ a.

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Forcing closed unbounded sets

Abstract: We discuss the problem of finding forcing posets which introduce closed unbounded subsets to a given stationary set.

Cited by 71 publications

References 10 publications

Adding many Baumgartner clubs

Adding many Baumgartner clubs

Definability degrees

A general Mitchell style iteration

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