The tree property at the double successor of a measurable cardinal κ with $2^{κ}$ large

Sy-David Friedman; Ajdin Halilović

Displaying similar documents to “The tree property at the double successor of a measurable cardinal κ with $2^{κ}$ large”

Supercompactness and failures of GCH

Sy-David Friedman, Radek Honzik (2012)

Fundamenta Mathematicae

Similarity:

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...

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

Laura Fontanella, Sy David Friedman (2015)

Fundamenta Mathematicae

Similarity:

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.

A partition property of cardinal numbers

N. H. Williams

Similarity:

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...

How many normal measures can $ℵ_{ω + 1}$ carry?

Arthur W. Apter (2006)

Fundamenta Mathematicae

Similarity:

We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for $ℵ_{ω + 1}$ to be measurable and to carry exactly τ normal measures, where $τ \geq ℵ_{ω + 2}$ is any regular cardinal. This contrasts with the fact that assuming AD + DC, $ℵ_{ω + 1}$ is measurable and carries exactly three normal measures. Our proof uses the methods of [6], along with a folklore technique and a new method due to James Cummings.

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

Lúcia Junqueira, Piotr Koszmider (2001)

Fundamenta Mathematicae

Similarity:

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...

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

Arthur W. Apter (2004)

Fundamenta Mathematicae

Similarity:

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.

Categoricity of theories in $L_{κ , ω}$ , when κ is a measurable cardinal. Part 2

Saharon Shelah (2001)

Fundamenta Mathematicae

Similarity:

We continue the work of [2] and prove that for λ successor, a λ-categorical theory T in $L_{κ *, ω}$ is μ-categorical for every μ ≤ λ which is above the $(2^{L S (T)}) ⁺$ -beth cardinal.

Iterating along a Prikry sequence

Spencer Unger (2016)

Fundamenta Mathematicae

Similarity:

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 $2^{ℵ_{ω}} = ℵ_{ω + 2}$ .

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

Stavros Christodoulou (1999)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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 $p \in X$ .

Reflecting character and pseudocharacter

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

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

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

Athanassios Tzouvaras (2004)

Fundamenta Mathematicae

Similarity:

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...

L-like Combinatorial Principles and Level by Level Equivalence

Arthur W. Apter (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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...

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

Asaf Karagila (2014)

Fundamenta Mathematicae

Similarity:

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....

Interpolation of $κ$ -compactness and PCF

István Juhász, Zoltán Szentmiklóssy (2009)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

Two cases when $d_{κ}$ and $d *_{κ}$ are equal

Ofer Shafir (2001)

Fundamenta Mathematicae

Similarity:

We deal with two cardinal invariants and give conditions on their equality using Shelah's pcf theory.

The instability of nonseparable complete Erdős spaces and representations in ℝ-trees

Jan J. Dijkstra, Kirsten I. S. Valkenburg (2010)

Fundamenta Mathematicae

Similarity:

One way to generalize complete Erdős space $_{c}$ is to consider uncountable products of zero-dimensional $G_{δ}$ -subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise...

Displaying similar documents to “The tree property at the double successor of a measurable cardinal κ with 2 κ large”

Displaying similar documents to “The tree property at the double successor of a measurable cardinal κ with $2^{κ}$ large”