CH and the Sacks property

S. Quickert

Displaying similar documents to “CH and the Sacks property”

Another ⋄-like principle

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 $\neg ⋄_{}$ 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...

Uncountable cardinals have the same monadic ∀₁¹ positive theory over large sets

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 $(2^{κ}, {[2^{κ}]}^{> κ}, <)$ , $(2^{λ}, {[2^{λ}]}^{> λ}, <)$ are...

Characterizing the powerset by a complete (Scott) sentence

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 $ϕ_{ℳ}$ if $ϕ_{ℳ}$ 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 $ℵ_{β}$ is characterized by a Scott sentence, then $2^{ℵ_{β + β ₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 <...

Locally Σ₁-definable well-orders of H(κ⁺)

Peter Holy, Philipp Lücke (2014)

Fundamenta Mathematicae

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Given an uncountable cardinal κ with $κ = κ^{< κ}$ and $2^{κ}$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^{κ}$ 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...

On Dimensionsgrad, resolutions, and chainable continua

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 $S_{n, α, β}$ such that (a) $d i m S_{n, α, β} = n$ ; (b) $t r D g S_{n, α, β} = t r D g o S_{n, α, β} = α$ ; (c) $t r i n d S_{n, α, β} = t r I n d ₀ S_{n, α, β} = β$ ; (d) if β < ω(⁺), then $S_{n, α, β}$ is separable and first countable; (e) if n = 1, then $S_{n, α, β}$ can be made chainable or hereditarily decomposable; (f) if α = β < ω(⁺), then $S_{n, α, β}$ can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(⁺), then $S_{n, α, β}$ can be made chainable and hereditarily indecomposable. In...

Iterating along a Prikry sequence

Spencer Unger (2016)

Fundamenta Mathematicae

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We introduce a new method which combines Prikry forcing with an iteration between the Prikry points. Using our method we prove from large cardinals that it is consistent that the tree property holds at ℵₙ for n ≥ 2, $ℵ_{ω}$ is strong limit and $2^{ℵ_{ω}} = ℵ_{ω + 2}$ .

On the bounding, splitting, and distributivity numbers

Alan S. Dow, Saharon Shelah (2023)

Commentationes Mathematicae Universitatis Carolinae

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The cardinal invariants $𝔥, 𝔟, 𝔰$ of $𝒫 (ω)$ are known to satisfy that $ω_{1} \leq 𝔥 \leq min {𝔟, 𝔰}$ . We prove that all inequalities can be strict. We also introduce a new upper bound for $𝔥$ and show that it can be less than $𝔰$ . 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).

Reflecting character and pseudocharacter

Lucia R. Junqueira, Alberto M. E. Levi (2015)

Commentationes Mathematicae Universitatis Carolinae

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We say that a cardinal function $φ$ reflects an infinite cardinal $κ$ , if given a topological space $X$ with $φ (X) \geq κ$ , there exists $Y \in {[X]}^{\leq κ}$ with $φ (Y) \geq κ$ . 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...

Coloring ordinals by reals

Jörg Brendle, Sakaé Fuchino (2007)

Fundamenta Mathematicae

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We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of $C^{s} (κ)$ and $F^{s} (κ)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence...

Embedding orders into the cardinals with $D C_{κ}$

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 $Z F + D C_{< κ}$ for any regular κ. We use this theorem to show that for all κ, the assumption of $D C_{κ}$ 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....

On families of Lindelöf and related subspaces of $2^{ω ₁}$

Lúcia Junqueira, Piotr Koszmider (2001)

Fundamenta Mathematicae

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We consider the families of all subspaces of size ω₁ of $2^{ω ₁}$ (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 ${[X]}^{ω ₁}$ 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...

Covering Property Axiom $C P A_{c u b e}$ and its consequences

Krzysztof Ciesielski, Janusz Pawlikowski (2003)

Fundamenta Mathematicae

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We formulate a Covering Property Axiom $C P A_{c u b e}$ , which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨ f ₙ \in ℝ^{ℝ} ⟩_{n < ω}$ of Borel functions...

Preservation of properties of a map by forcing

Akira Iwasa (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $f : X \to Y$ be a continuous map such as an open map, a closed map or a quotient map. We study under what circumstances $f$ remains an open, closed or quotient map in forcing extensions.

On ordinals accessible by infinitary languages

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 $L_{λ ⁺ ω}$ , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with $⟨ D^{ℳ}, ≺^{ℳ} ⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where $⟨ D^{ℳ^{'}}, ≺^{ℳ^{'}} ⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{λ ⁺ ω}$ is exactly $ℶ_{δ (λ)}$ . It was proved in [BK71] that if ℵ₀ < λ < κ $a r e r e g u l a r c a r d i n a l n u m b e r s, t h e n t h e r e i s a f o r c i n g e x t e n s i o n, p r e s e r v i n g c o f i n a l i t i e s, s u c h t h a t i n t h e e x t e n s i o n$ 2λ = κ $a n d δ (λ) < λ ⁺ ⁺ . W e i m p r o v e t h i s r e s u l t b y p r o v i n g t h e f o l l o w i n g : S u p p o s e ℵ ₀ < λ < θ \leq κ a r e c a r d i n a l n u m b e r s s u c h t h a t$ ∙ $λ^{< λ} = λ$ ; ∙ cf(θ) ≥ λ⁺ and $μ^{λ} < θ$ whenever μ < θ; ∙ $κ^{λ} = κ$ . Then there...

Singular arc-like continua

Tadeusz Maćkowiak

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CONTENTSIntroduction.......................................................................................................51. Preliminaries.................................................................................................6 A. Mappings....................................................................................................6 B. Arc-like continua.........................................................................................8 C. Pseudosuspensions...................................................................................8 D....

A solution to Comfort's question on the countable compactness of powers of a topological group

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 $α \leq 2^{}$ , a topological group G such that $G^{γ}$ is countably compact for all cardinals γ < α, but $G^{α}$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M A_{c o u n t a b l e}$ . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M A_{c o u n t a b l e}$ . However, the question has...

A new Lindelöf space with points $G_{δ}$

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove that $♦^{*}$ implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality $2^{ℵ_{1}}$ which has points $G_{δ}$ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

A partition property of cardinal numbers

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...

The tree property at both $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$

Laura Fontanella, Sy David Friedman (2015)

Fundamenta Mathematicae

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We force from large cardinals a model of ZFC in which $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$ both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model $ℵ_{ω + 2}$ even satisfies the super tree property.

Counting models of set theory

Ali Enayat (2002)

Fundamenta Mathematicae

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Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and $κ \in ω \cup ℵ ₀, ℵ ₁, 2^{ℵ ₀}$ . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $μ (T, α) = 2^{ℵ ₀}$ . (3) If α < ω₁ and T ⊢ V ≠ OD,...

Existentially closed II₁ factors

Ilijas Farah, Isaac Goldbring, Bradd Hart, David Sherman (2016)

Fundamenta Mathematicae

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We examine the properties of existentially closed ( $^{ω}$ -embeddable) II₁ factors. In particular, we use the fact that every automorphism of an existentially closed ( $^{ω}$ -embeddable) II₁ factor is approximately inner to prove that Th() is not model-complete. We also show that Th() is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th().

A continuum $X$ such that $C (X)$ is not continuously homogeneous

Alejandro Illanes (2016)

Commentationes Mathematicae Universitatis Carolinae

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A metric continuum $X$ is said to be continuously homogeneous provided that for every two points $p, q \in X$ there exists a continuous surjective function $f : X \to X$ such that $f (p) = q$ . Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum $X$ such that the hyperspace of subcontinua of $X$ , $C (X)$ , is not continuously homogeneous.

Indestructible colourings and rainbow Ramsey theorems

Lajos Soukup (2009)

Fundamenta Mathematicae

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We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that $2^{ω ₁}$ is arbitrarily large,...

Decidability and definability results related to the elementary theory of ordinal multiplication

Alexis Bès (2002)

Fundamenta Mathematicae

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The elementary theory of ⟨α;×⟩, where α is an ordinal and × denotes ordinal multiplication, is decidable if and only if $α < ω^{ω}$ . Moreover if $|_{r}$ and $|_{l}$ respectively denote the right- and left-hand divisibility relation, we show that Th $⟨ ω^{ω^{ξ}} {; |}_{r} ⟩$ and Th $⟨ ω^{ξ} {; |}_{l} ⟩$ are decidable for every ordinal ξ. Further related definability results are also presented.

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

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We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $𝒮$ of morphisms in a locally presentable category $𝒞$ of structures, the orthogonal class of objects...

Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, William Weiss (2006)

Fundamenta Mathematicae

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Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put $_{λ} (α) = s \in (α) : s (0) = λ = m i n [s (β) : β < α]$ . We show that f ∈ (α) iff for some natural number n there are infinite cardinals $λ ₀ i > λ ₁ > . . . > λ_{n - 1}$ and ordinals $α ₀, . . ., α_{n - 1}$ such that $α = α ₀ + \dots + α_{n - 1}$ and $f = f ₀ ⏜ f ₁ ⏜ . . . ⏜ f_{n - 1}$ where each $f_{i} \in_{λ_{i}} (α_{i})$ . Under GCH we prove that if α < ω₂ then (i) $_{ω} (α) = s \in^{α} ω, ω ₁ : s (0) = ω$ ; (ii) if λ > cf(λ) = ω, $_{λ} (α) = s \in^{α} λ, λ ⁺ : s (0) = λ, s^{- 1} λ i s ω ₁ - c l o s e d i n α$ ; (iii) if cf(λ) = ω₁, $_{λ} (α) = s \in^{α} λ, λ ⁺ : s (0) = λ, s^{- 1} λ i s ω - c l o s e d a n d s u c c e s s o r - c l o s e d i n α$ ; (iv) if cf(λ) > ω₁, $_{λ} (α) =^{α} λ$ . This yields a complete characterization of the classes (α) for all...

Internally club and approachable for larger structures

John Krueger (2008)

Fundamenta Mathematicae

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We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of $P_{κ} (X)$ , where κ ⊆ X. Suppose μ < κ, $μ^{< μ} = μ$ , and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in ${[H (λ)]}^{μ ⁺}$ which are internally club but not internally approachable.

On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product $2^{ℝ}$

Eleftherios Tachtsis (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets ( $A C^{W O}$ ) does not imply “the Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is countably compact” in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets...

Partition ideals below $ℵ_{ω}$

P. Dodos, J. Lopez-Abad, S. Todorcevic (2012)

Fundamenta Mathematicae

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Motivated by an application to the unconditional basic sequence problem appearing in our previous paper, we introduce analogues of the Laver ideal on ℵ₂ living on index sets of the form ${[ℵ_{k}]}^{ω}$ and use this to refine the well-known high-dimensional polarized partition relation for $ℵ_{ω}$ of Shelah.

Some remarks on universality properties of $ℓ_{\infty} / c ₀$

Mikołaj Krupski, Witold Marciszewski (2012)

Colloquium Mathematicae

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We prove that if is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into $ℓ_{\infty} / c ₀$ . We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into $ℓ_{\infty} / c ₀$ , but fails to embed...