On the consistency of some partition theorems for continuous colorings, and the structure of ℵ1-dense real order types

Abraham, Uri; Rubin, Matatyahu; Shelah, Saharon

doi:10.1016/0168-0072(84)90024-1

Cited by 96 publications

(117 citation statements)

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“…Thus OCA is a special case of OCA ∞ , when K n 0 = K 0 for all n. In [9, §3] it was proved that PFA implies OCA ∞ , as (a) from [9, §3] implies (1) and (b') of [9, §3] implies (2). It is not known whether OCA and OCA ∞ are equivalent statements.…”

Section: Oca If X Is a Separable Metric Space And [X]mentioning

confidence: 99%

Luzin gaps

Farah

2004

Trans. Amer. Math. Soc.

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Abstract. We isolate a class of F σδ ideals on N that includes all analytic P-ideals and all Fσ ideals, and introduce 'Luzin gaps' in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author's rigidity conjecture for quotients P(N)/I. We also prove that the ideals NWD(Q) and NULL(Q) have the Radon-Nikodým property, and (using OCA∞) a uniformization result for K-coherent families of continuous partial functions.One of the most fascinating facts about the Boolean algebra P(N)/ Fin was discovered by Hausdorff in 1908. In [20], he constructed two families A and B of sets of integers such that (a) A ∩ B is finite for all A ∈ A and all B ∈ B, (b) for every C ⊆ N either A \ C is infinite for some A ∈ A or B ∩ C is infinite for some B ∈ B, and (c) both and A and B have order-type equal to ω 1 with respect to the inclusion modulo finite. Families A and B satisfying (a) are orthogonal, those satisfying both (a) and (b) form a gap in P(N)/ Fin (or, they are not separated over Fin, the ideal of finite sets), and if they furthermore satisfy (c) they form an (ω 1 , ω 1 )-gap. If (a) and (b) hold and both A and B are linearly ordered by the inclusion modulo finite, a gap is linear. A gap (linear or not) is Hausdorff if both of its sides are σ-directed under the inclusion modulo finite.Another major advance in the study of gaps in P(N)/ Fin was made by Kunen ([29]), who used a condition originally isolated by Luzin in [32] to prove that for every (ω 1 , ω 1 )-gap there is a ccc poset that freezes it. (A gap is indestructible, or frozen, if it remains a gap in every ℵ 1 -preserving forcing extension.) Remarkably, a linear gap can be frozen by an ℵ 1 -preserving forcing if and only if it contains an (ω 1 , ω 1 )-gap. Kunen used freezing to prove that P(N)/ Fin need not be universal for linear orderings of size at most 2 ℵ0 even if Martin's Axiom, MA, is assumed. The technique of freezing gaps has played an important role in a variety of subjects, from automatic continuity in Banach algebras ([6]) to the study of automorphisms of P(N)/ Fin ([35]).

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Section: Oca If X Is a Separable Metric Space And [X]mentioning

confidence: 99%

Luzin gaps

Farah

2004

Trans. Amer. Math. Soc.

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“…Clearly, stronger axioms than MM could not be invented, or at least not in an obvious way, and hence research concentrated for some years on studying the weakenings of these axioms, especially weakenings of PFA. Some of the popular choices are OCA, the open colouring axiom (it has two versions, introduced respectively by Abraham, Rubin and Shelah in [1] and Todorčević in [33]), BPFA, which is the bounded PFA, introduced by Goldstern and Shelah in [18], MPR introduced by Moore in [24] and, of particular interest, PID, the P -ideal dichotomy which has the surprising feature to be consistent with the continuum hypothesis, as shown by Abraham and Todorčević in [2]. The interest of these axioms is exactly in their relative weakness, as they allow us to get the relative consistency of statements that are seemingly contradictory, such as CH and certain consequences of PID.…”

Section: Away From Propernessmentioning

confidence: 99%

“…A set of conditions is called an antichain if it consists of pairwise incompatible conditions. 1 The moral opposite of an antichain is a filter, which is a set in which every two conditions are compatible, moreover with a common extension in the filter itself. We also assume that filters are closed under weakenings of the conditions within the filter.…”

Section: The Discovery Of Forcingmentioning

confidence: 99%

Forcing Axioms, Finite Conditions and Some More

Džamonja

2013

Logic and Its Applications

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Abstract. We survey some classical and some recent results in the theory of forcing axioms, aiming to present recent breakthroughs and interest the reader in further developing the theory. The article is written for an audience of logicians and mathematicians not necessarily familiar with set theory. IntroductionWe shall work within the axioms of the Zermelo-Fraenkel set theory with Choice (ZFC). These axioms were introduced basically starting from 1908 and improving to a final version in the 1920s as an attempt to axiomatize the foundations of mathematics. There have been other such attempts at about the same time and later, but it is fair to say that for the purposes of much of modern mathematics the axioms of ZFC represent the accepted foundation (see [13] for a detailed discussion of foundational issues in set theory). Gödel's Incompleteness theorems [16] prove that for any consistent theory T which implies the Peano Axioms and whose axioms are presentable as a recursively enumerable set of sentences, so for any reasonable theory one would say, there is a sentence ϕ in the language of T such that T does not prove or disprove ϕ. In some sense the discussion of which axioms to use is made less interesting by these theorems, which can be interpreted as saying that a perfect choice of axioms does not exists. We therefore do like the most, we concentrate on the axioms that correctly model most of mathematics, and for the rest, we try to understand the limits and how we can improve them. For us ZFC is a basis for a foundation which in some circumstances can be extended to a larger set of axioms which provide an insight into various parts of mathematics. In here we concentrate on the forcing axioms (and their negations).

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“…Let {A i } i∈k be open balls centred at points of p such that the sum of the diameters of the balls A i is less than (1 − δ)/3. If (c 0 , {a 0 i } i∈k , 0 ) and (c 1 , {a 1 i } i∈k , 1 ) are such that a j i ∈ A i , j > 4 and c j − p < , then (c 0 , {a 0 i } i∈k , 0 ) and (c 1 , {a 1 i } i∈k , 1 ) are compatible. Trying to mimic this to obtain a smooth decomposition would entail adding the derivative to the condition (c, x, ) and insisting that extension now means that derivatives stay within of each other.…”

Section: Introductionmentioning

confidence: 99%

“…For example, it is easy to use σ-linked 1 forcing to decompose any Euclidean space into ℵ 1 continuous, oneto-one and even rectifiable 2 curves. To see that this is not so easily extended to differentiable curves, consider the natural forcing for decomposing the plane into rectifiable curves.…”

Section: Introductionmentioning

confidence: 99%

Decomposing Euclidean space with a small number of smooth sets

Steprāns

1999

Trans. Amer. Math. Soc.

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Abstract. Let the cardinal invariant sn denote the least number of continuously smooth n-dimensional surfaces into which (n + 1)-dimensional Euclidean space can be decomposed. It will be shown to be consistent that sn is greater than s n+1 . These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.

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On the consistency of some partition theorems for continuous colorings, and the structure of ℵ1-dense real order types

Cited by 96 publications

References 9 publications

Luzin gaps

Luzin gaps

Forcing Axioms, Finite Conditions and Some More

Decomposing Euclidean space with a small number of smooth sets

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