Reflection implies the SCH

Saharon Shelah

Displaying similar documents to “Reflection implies the SCH”

Some properties of stationary sets

C. A. Di Prisco, W. Marek

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CONTENTSIntroduction..................................................................51. Derivative of a stationary set...................................72. Stationary degrees ...............................................133. Subsets of $P_{ϰ} (λ)$ ..............................................194. Stationary subsets of $P_{ϰ} (λ)$ .............................255. Superstationary substes of $P_{ϰ} (λ)$ ....................326. End-stationary subsets of $P_{ϰ} (λ)$ ......................34References................................................................37 ...

Splitting stationary sets in $_{κ} λ$ for λ with small cofinality

Toshimichi Usuba (2009)

Fundamenta Mathematicae

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For a regular uncountable cardinal κ and a cardinal λ with cf(λ) < κ < λ, we investigate the consistency strength of the existence of a stationary set in $_{κ} λ$ which cannot be split into λ⁺ many pairwise disjoint stationary subsets. To do this, we introduce a new notion for ideals, which is a variation of normality of ideals. We also prove that there is a stationary set S in $_{κ} λ$ such that every stationary subset of S can be split into λ⁺ many pairwise disjoint stationary subsets. ...

Interpolation of $κ$ -compactness and PCF

István Juhász, Zoltán Szentmiklóssy (2009)

Commentationes Mathematicae Universitatis Carolinae

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We call a topological space $κ$ -compact if every subset of size $κ$ has a complete accumulation point in it. Let $Φ (μ, κ, λ)$ denote the following statement: $μ < κ < λ = cf (λ)$ and there is ${S_{ξ} : ξ < λ} \subset {[κ]}^{μ}$ such that $| {ξ : | S_{ξ} \cap A | = μ} | < λ$ whenever $A \in {[κ]}^{< κ}$ . We show that if $Φ (μ, κ, λ)$ holds and the space $X$ is both $μ$ -compact and $λ$ -compact then $X$ is $κ$ -compact as well. Moreover, from PCF theory we deduce $Φ (cf (κ), κ, κ^{+})$ for every singular cardinal $κ$ . As a corollary we get that a linearly Lindelöf and $ℵ_{ω}$ -compact space is uncountably compact, that is $κ$ -compact for all uncountable cardinals...

Δ₁-Definability of the non-stationary ideal at successor cardinals

Sy-David Friedman, Liuzhen Wu, Lyubomyr Zdomskyy (2015)

Fundamenta Mathematicae

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Assuming V = L, for every successor cardinal κ we construct a GCH and cardinal preserving forcing poset ℙ ∈ L such that in $L^{ℙ}$ the ideal of all non-stationary subsets of κ is Δ₁-definable over H(κ⁺).

Supercompactness and partial level by level equivalence between strong compactness and strongness

Arthur W. Apter (2004)

Fundamenta Mathematicae

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We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if $δ^{+ α}$ is regular, δ is $δ^{+ α}$ strongly compact iff δ is δ + α + 1 strong, except possibly if δ is a limit of cardinals γ which are $δ^{+ α}$ strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible.

A Note on Indestructibility and Strong Compactness

Arthur W. Apter (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is $2^{λ}$ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is $2^{δ} = δ ⁺$ supercompact,...

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, R. L. Schilling (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$ . The state space of $(Z, S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z, S)$ is the product of the uniform probability measure and a Gaussian distribution.

L-like Combinatorial Principles and Level by Level Equivalence

Arthur W. Apter (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like” combinatorial principles. In particular, this model satisfies the following properties: (1) $♢_{δ}$ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ₀ such that for every δ ∈ S, $◻_{δ}$ holds and δ carries a gap 1 morass. (3) A weak version of $◻_{δ}$ holds for every...

On reduction of two-parameter prediction problems

J. Friedrich, L. Klotz, M. Riedel (1995)

Studia Mathematica

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We present a general method for the extension of results about linear prediction for q-variate weakly stationary processes on a separable locally compact abelian group $G_{2}$ (whose dual is a Polish space) with known values of the processes on a separable subset $S_{2} \subseteq G_{2}$ to results for weakly stationary processes on $G_{1} \times G_{2}$ with observed values on $G_{1} \times S_{2}$ . In particular, the method is applied to obtain new proofs of some well-known results of Ze Pei Jiang.

Initially $κ$ -compact spaces for large $κ$

Stavros Christodoulou (1999)

Commentationes Mathematicae Universitatis Carolinae

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This work presents some cardinal inequalities in which appears the closed pseudo-character, $ψ_{c}$ , of a space. Using one of them — $ψ_{c} (X) \leq 2^{d (X)}$ for $T_{2}$ spaces — we improve, from $T_{3}$ to $T_{2}$ spaces, the well-known result that initially $κ$ -compact $T_{3}$ spaces are $λ$ -bounded for all cardinals $λ$ such that $2^{λ} \leq κ$ . And then, using an idea of A. Dow, we prove that initially $κ$ -compact $T_{2}$ spaces are in fact compact for $κ = 2^{F (X)}$ , $2^{s (X)}$ , $2^{t (X)}$ , $2^{χ (X)}$ , $2^{ψ_{c} (X)}$ or $κ = max {τ^{+}, τ^{< τ}}$ , where $τ > t (p, X)$ for all $p \in X$ .

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...

Guessing clubs in the generalized club filter

Bernhard König, Paul Larson, Yasuo Yoshinobu (2007)

Fundamenta Mathematicae

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We present principles for guessing clubs in the generalized club filter on $_{κ} λ$ . These principles are shown to be weaker than classical diamond principles but often serve as sufficient substitutes. One application is a new construction of a λ⁺-Suslin-tree using assumptions different from previous constructions. The other application partly solves open problems regarding the cofinality of reflection points for stationary subsets of ${[λ]}^{ℵ ₀}$ .

Separating by $G_{δ}$ -sets in finite powers of ω₁

Yasushi Hirata, Nobuyuki Kemoto (2003)

Fundamenta Mathematicae

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It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint $G_{δ}$ -sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.

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...

Supercompactness and failures of GCH

Sy-David Friedman, Radek Honzik (2012)

Fundamenta Mathematicae

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Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s...

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...

The nonexistence of robust codes for subsets of ω₁

David Asperó (2005)

Fundamenta Mathematicae

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Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence $⟨ S_{α} : α < ω ₁ ⟩$ of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that $S \in L [r, ⟨ S *_{α} : α < ω ₁ ⟩]$ whenever each $S *_{α}$ is equal to $S_{α}$ modulo nonstationary changes, or may have the weaker meaning that $S \in L [r, ⟨ S_{α} \cap C : α < ω ₁ ⟩]$ for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that...

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...

Banach-space-valued stationary processes and their linear prediction

S. A. Chobanyan, A. Weron

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Contents0. Introduction............................................................................................................................................. 51. Linear operators generated by random elements.......................................................................... 62. Covariance operator of generalized random elements................................................................. 93. The space of generalized random elements of the second-order as an LVH-space.................