CH and the Sacks property
S. Quickert (2002)
Fundamenta Mathematicae
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We show the consistency of CH and the statement “no ccc forcing has the Sacks property” and derive some consequences for ccc -bounding forcing notions.
S. Quickert (2002)
Fundamenta Mathematicae
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We show the consistency of CH and the statement “no ccc forcing has the Sacks property” and derive some consequences for ccc -bounding forcing notions.
Michael Hrušák (2001)
Fundamenta Mathematicae
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A new ⋄-like principle consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that is consistent with CH and that in many models of = ω₁ the principle holds. As implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁...
Michael G. Charalambous, Jerzy Krzempek (2010)
Fundamenta Mathematicae
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For each natural number n ≥ 1 and each pair of ordinals α,β with n ≤ α ≤ β ≤ ω(⁺), where ω(⁺) is the first ordinal of cardinality ⁺, we construct a continuum such that (a) ; (b) ; (c) ; (d) if β < ω(⁺), then is separable and first countable; (e) if n = 1, then can be made chainable or hereditarily decomposable; (f) if α = β < ω(⁺), then can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(⁺), then can be made chainable and hereditarily indecomposable. In...
Lucia R. Junqueira, Alberto M. E. Levi (2015)
Commentationes Mathematicae Universitatis Carolinae
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We say that a cardinal function reflects an infinite cardinal , if given a topological space with , there exists with . We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences...
Stavros Christodoulou (1999)
Commentationes Mathematicae Universitatis Carolinae
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This work presents some cardinal inequalities in which appears the closed pseudo-character, , of a space. Using one of them — for spaces — we improve, from to spaces, the well-known result that initially -compact spaces are -bounded for all cardinals such that . And then, using an idea of A. Dow, we prove that initially -compact spaces are in fact compact for , , , , or , where for all .
Alan S. Dow, Saharon Shelah (2023)
Commentationes Mathematicae Universitatis Carolinae
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The cardinal invariants of are known to satisfy that . We prove that all inequalities can be strict. We also introduce a new upper bound for and show that it can be less than . The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).
Ljubiša D. R. Kočinac, Marion Scheepers (2003)
Fundamenta Mathematicae
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We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a -space. In [9] we showed that has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. has countable fan tightness and the Reznichenko...
Lúcia Junqueira, Piotr Koszmider (2001)
Fundamenta Mathematicae
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We consider the families of all subspaces of size ω₁ of (or of a compact zero-dimensional space X of weight ω₁ in general) which are normal, have the Lindelöf property or are closed under limits of convergent ω₁-sequences. Various relations among these families modulo the club filter in are shown to be consistently possible. One of the main tools is dealing with a subspace of the form X ∩ M for an elementary submodel M of size ω₁. Various results with this flavor are obtained. Another...
Artur Hideyuki Tomita (2005)
Fundamenta Mathematicae
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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number , a topological group G such that is countably compact for all cardinals γ < α, but is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from . However, the question has...
Athanassios Tzouvaras (2004)
Fundamenta Mathematicae
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We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form (∀X)ϕ(X) and (∃X)ϕ(X), for ϕ positive in X and containing no set-quantifiers, when the set variables range over large (= cofinal) subsets of the cardinals. This strengthens the result of Doner-Mostowski-Tarski [3] that (κ,∈), (λ,∈) are elementarily equivalent when κ, λ are uncountable. It follows that we can consistently postulate that the structures , are...
Asaf Karagila (2014)
Fundamenta Mathematicae
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Jech proved that every partially ordered set can be embedded into the cardinals of some model of ZF. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of for any regular κ. We use this theorem to show that for all κ, the assumption of does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model....
Pierre Matet (2002)
Fundamenta Mathematicae
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We show that cov(M) is the least infinite cardinal λ such that (the set of all finite subsets of λ ) fails to satisfy a certain natural generalization of Ramsey’s Theorem.
N. H. Williams
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CONTENTSIntroduction....................................................................................... 5§ 1. Notation and definitions......................................................... 5§ 2. Negative relations.................................................................... 9§ 3. The Ramification Lemma ..................................................... 10§ 4. The main theorem................................................................... 13§ 5. A result for cardinals...
Ioannis Souldatos (2013)
Fundamenta Mathematicae
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This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence if has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if is characterized by a Scott sentence, then is (homogeneously) characterized by a Scott sentence, for all 0 <...
Saharon Shelah, Pauli Väisänen, Jouko Väänänen (2005)
Fundamenta Mathematicae
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Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where is non-well-ordered. One of the interesting properties of this number is that the Hanf number of is exactly . It was proved in [BK71] that if ℵ₀ < λ < κ2λ = κ∙ ; ∙ cf(θ) ≥ λ⁺ and whenever μ < θ; ∙ . Then there...
Peter Holy, Philipp Lücke (2014)
Fundamenta Mathematicae
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Given an uncountable cardinal κ with and regular, we show that there is a forcing that preserves cofinalities less than or equal to and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein...