scholarly journals Infinite Monochromatic Paths and a Theorem of Erdős-Hajnal-Rado

10.37236/8849 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Shimon Garti ◽  
Menachem Magidor ◽  
Saharon Shelah
Keyword(s):  
Singular Cardinal

We prove that if $\mu$ is a singular cardinal with countable cofinality and $2^\mu=\mu^+$ then $\binom{\mu^+}{\mu}\nrightarrow\binom{\mu^+\ \aleph_2}{\mu\ \mu}$.

scholarly journals Exact saturation in simple and NIP theories

2017 ◽  
Vol 17 (01) ◽  
pp. 1750001 ◽  
Author(s):  
Itay Kaplan ◽  
Saharon Shelah ◽  
Pierre Simon
Keyword(s):  
Simple Theory ◽  
Saturated Model ◽  
Singular Cardinal ◽  
Theory Formula

A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: see text]-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.

scholarly journals The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2

2012 ◽  
Vol 77 (3) ◽  
pp. 934-946 ◽  
Author(s):  
Dima Sinapova
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis

AbstractWe show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵω2+ 1 and the SCH fails at ℵω2.

scholarly journals Contributions to the Theory of Large Cardinals through the Method of Forcing

2021 ◽  
Vol 27 (2) ◽  
pp. 221-222
Author(s):  
Alejandro Poveda
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Tree Property ◽  
Prikry Forcing ◽  
Regular Cardinals ◽  
Singular Cardinals ◽  
Vopěnka’S Principle

AbstractThe dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle (Part I). In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics (Part II and Part III).We commence Part I by investigating the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{(n)}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $(\lambda ^{+\omega })^{\mathrm {HOD}}<\lambda ^+$ , for every regular cardinal $\lambda $ .” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $(\lambda ^+)^{\textrm {{HOD}}}=\lambda ^+$ , there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$ , whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation (Part III) we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously.

THE TREE PROPERTY AT THE TWO IMMEDIATE SUCCESSORS OF A SINGULAR CARDINAL

2020 ◽  
pp. 1-9
Author(s):  
JAMES CUMMINGS ◽  
YAIR HAYUT ◽  
MENACHEM MAGIDOR ◽  
ITAY NEEMAN ◽  
DIMA SINAPOVA ◽  
...  
Keyword(s):  
Singular Cardinal ◽  
Tree Property

On inverse γ-systems and the number of L∞λ-equivalent, non-isomorphic models for λ singular

2000 ◽  
Vol 65 (1) ◽  
pp. 272-284
Author(s):  
Saharon Shelah ◽  
Pauli Väisänen
Keyword(s):  
Abelian Groups ◽  
Singular Cardinal

AbstractSuppose λ is a singular cardinal of uncountable cofinality κ. For a model of cardinality λ, let No() denote the number of isomorphism types of models of cardinality λ which are L∞λ-equivalent to . In [7] Shelah considered inverse κ-systems of abelian groups and their certain kind of quotient limits Gr()/ Fact(). In particular Shelah proved in [7, Fact 3.10] that for every cardinal Μ there exists an inverse κ-system such that consists of abelian groups having cardinality at most Μκ and card(Gr()/ Fact()) = Μ. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that θκ < λ for every θ < λ): if is an inverse κ-system of abelian groups having cardinality < λ, then there is a model such that card() = λ and No() = card(Gr()/ Fact()). The following was an immediate consequence (when θκ < λ for every θ < λ): for every nonzero Μ < λ or Μ = λκ there is a model , of cardinality λ with No() = Μ. In this paper we show: for every nonzero Μ ≤ λκ there is an inverse κ-system of abelian groups having cardinality < λ such that card(Gr()/ Fact()) = Μ (under the assumptions 2κ < λ and θ<κ < λ for all θ < λ when Μ > λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No() = λ is possible when θκ > λ for every λ < λ.

A definable failure of the singular cardinal hypothesis

2012 ◽  
Vol 192 (2) ◽  
pp. 719-762 ◽  
Author(s):  
Sy-David Friedman ◽  
Radek Honzik
Keyword(s):  
Singular Cardinal ◽  
Singular Cardinal Hypothesis

scholarly journals ARONSZAJN TREES AND FAILURE OF THE SINGULAR CARDINAL HYPOTHESIS

2009 ◽  
Vol 09 (01) ◽  
pp. 139-157 ◽  
Author(s):  
ITAY NEEMAN
Keyword(s):  
Large Cardinals ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Tree Property ◽  
Singular Cardinal Hypothesis ◽  
Aronszajn Trees ◽  
Strongly Compact Cardinals

The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ.

NOWHERE PRECIPITOUSNESS OF THE NON-STATIONARY IDEAL OVER ${\mathcal P}_\kappa \lambda$

2002 ◽  
Vol 02 (01) ◽  
pp. 81-89 ◽  
Author(s):  
YO MATSUBARA ◽  
SAHARON SHELAH
Keyword(s):  
Stationary Subset ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Uncountable Cardinal

We prove that if λ is a strong limit singular cardinal and κ a regular uncountable cardinal < λ, then NSκλ, the non-stationary ideal over [Formula: see text], is nowhere precipitous. We also show that under the same hypothesis every stationary subset of [Formula: see text] can be partitioned into λκ disjoint stationary sets.

scholarly journals Diagonal Prikry extensions

2010 ◽  
Vol 75 (4) ◽  
pp. 1383-1402 ◽  
Author(s):  
James Cummings ◽  
Matthew Foreman
Keyword(s):  
Model Theory ◽  
Measurable Cardinal ◽  
Core Model ◽  
Singular Cardinal ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Singular Cardinals ◽  
Free Abelian Groups ◽  
Woodin Cardinal

§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.Strong covering implies weak covering.In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include1. X is “0# exists”, Kx is L, Y is “strongly”.2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

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