Martin’s axiom does not imply that every two ℕ1-dense sets of reals are isomorphic

Avraham, Uri; Shelah, Saharon

doi:10.1007/bf02761858

Cited by 52 publications

(42 citation statements)

References 3 publications

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“…This shows (see also 2.6 in [31] and §5 of [1]) that the notion of (f ξ ) ξ<κ -entangled sets make sense for arbitrary split compact space even in a nonsymmetric setting (i.e., when I i = J i . )…”

Section: Totally Disconnected Nonreflection In All Continuous Imagesmentioning

confidence: 77%

“…Consistent examples providing positive answer to Question 1.1 (1) have been well known, for example, assuming the continuum hypothesis (CH), V. Fedorchuk showed that there are compact spaces where every 2010 Mathematics Subject Classification. 54D30, 03E35, 46B25.…”

Section: That L Is Nonmetrizable and Totally Disconnected?mentioning

confidence: 99%

“…The paper can be summarized as an attempt to construct spaces providing negative answers to Questions 1.1 (1) and (2) Our approach of considering split compact spaces as in Definition 2.1 to attack Question 1.1 (1) turns out to be successful. Our ZFC example from Theorem 3.2 which answers Question 1.1 (1) is based on a simple combinatorial principle (Lemma 3.1) discovered by K. Ciesielski and R. Pol in [6] as Remark 7.1 (see also [20]).…”

Section: That L Is Nonmetrizable and Totally Disconnected?mentioning

confidence: 99%

“…It can be described in the language similar to our unordered split interval: define an inverse limit system K α ⊆ [0, 1] α with α ≤ ω 1 containing the point 0 α as a nonisolated point. Given 1] for any neighbourhood of 0 α . The obtained K = K ω1 contains points of the form x r for r ∈ (0, 1] and 0 α⌢ x r for r ∈ (0, 1] and point 0 ω1 .…”

Section: Totally Disconnected Nonreflection In All Continuous Imagesmentioning

confidence: 99%

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On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider

2016

Topology and its Applications

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Abstract. We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L ⊆ K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open.

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Section: Totally Disconnected Nonreflection In All Continuous Imagesmentioning

confidence: 77%

Section: That L Is Nonmetrizable and Totally Disconnected?mentioning

confidence: 99%

Section: That L Is Nonmetrizable and Totally Disconnected?mentioning

confidence: 99%

Section: Totally Disconnected Nonreflection In All Continuous Imagesmentioning

confidence: 99%

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On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider

2016

Topology and its Applications

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“…Baumgartner proved in [4] that it is consistent that every two such sets are order isomorphic and CH fails. Yet, Abraham and Shelah [3] proved that this statement does not follow from MA+¬ CH. Baumgartner's proof uses PFA, the proper forcing axiom.…”

Section: Beyond CCCmentioning

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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