Getting more colors II

2013 ◽  
Vol 78 (1) ◽  
pp. 17-38 ◽  
Author(s):  
Todd Eisworth
Keyword(s):  
Singular Cardinals ◽  
Coloring Theorem

AbstractWe formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle Pr1(μ+,μ+,μ+, cf (μ)) for singularμ.

Getting more colors I

2013 ◽  
Vol 78 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Todd Eisworth
Keyword(s):  
Singular Cardinals ◽  
Coloring Theorem

AbstractWe establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinalμ, we haveif and only iffor arbitrarily largeθ<μ.

scholarly journals The combinatorics of combinatorial coding by a real

1995 ◽  
Vol 60 (1) ◽  
pp. 36-57
Author(s):  
Saharon Shelah ◽  
Lee J. Stanley
Keyword(s):  
Singular Cardinals ◽  
Essential Properties

AbstractWe lay the combinatorial foundations for [5] by setting up and proving the essential properties of the coding apparatus for singular cardinals. We also prove another result concerning the coding apparatus for inaccessible cardinals.

Homogeneous changes in cofinalities with applications to HOD

2017 ◽  
Vol 17 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Omer Ben-Neria ◽  
Spencer Unger
Keyword(s):  
Large Cardinals ◽  
Related Question ◽  
New Technique ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
A New Technique

We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [Formula: see text] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [Formula: see text] in [Formula: see text] is [Formula: see text]-supercompact in HOD.

An AD-like model

1985 ◽  
Vol 50 (2) ◽  
pp. 531-543 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinals ◽  
Strong Sense ◽  
Supercompact Cardinals ◽  
Singular Cardinals ◽  
Partial Orderings ◽  
Measurable Cardinals ◽  
Jonsson Cardinals

A very fruitful line of research in recent years has been the application of techniques in large cardinals and forcing to the production of models in which certain consequences of the axiom of determinateness (AD) are true or in which certain “AD-like” consequences are true. Numerous results have been published on this subject, among them the papers of Bull and Kleinberg [4], Bull [3], Woodin [15], Mitchell [11], and [1], [2].Another such model will be constructed in this paper. Specifically, the following theorem is proven.Theorem 1. Con(ZFC + There are cardinals κ < δ < λ so that κ is a supercompact limit of supercompact cardinals, λ is a measurable cardinal, and δ is λ supercompact) ⇒ Con(ZF + ℵ1 and ℵ2 are Ramsey cardinals + The ℵn for 3 ≤ n ≤ ω are singular cardinals of cofinality ω each of which carries a Rowbottom filter + ℵω + 1 is a Ramsey cardinal + ℵω + 2 is a measurable cardinal).It is well known that under AD + DC, ℵ2 and ℵ2 are measurable cardinals, the ℵn for 3 ≤ n < ω are singular Jonsson cardinals of cofinality ℵ2, ℵω is a Rowbottom cardinal, and ℵω + 1 and ℵω + 2 are measurable cardinals.The proof of the above theorem will use the existence of normal ultrafilters which satisfy a certain property (*) (to be defined later) and an automorphism argument which draws upon the techniques developed in [9], [2], and [4] but which shows in addition that certain supercompact Prikry partial orderings are in a strong sense “homogeneous”. Before beginning the proof of the theorem, however, we briefly mention some preliminaries.

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.

scholarly journals THE STRONG TREE PROPERTY AT SUCCESSORS OF SINGULAR CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 193-207 ◽  
Author(s):  
LAURA FONTANELLA
Keyword(s):  
Singular Limit ◽  
Inaccessible Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinals ◽  
Image Position ◽  
Strongly Compact Cardinals

Abstract An inaccessible cardinal is strongly compact if, and only if, it satisfies the strong tree property. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where ${\aleph _{\omega + 1}}$ has the strong tree property. Moreover, we prove that every successor of a singular limit of strongly compact cardinals has the strong tree property.

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