On models with power-like orderings

1972 ◽  
Vol 37 (2) ◽  
pp. 247-267 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Singular Cardinal ◽  
Strong Limit

We prove here theorems of the form: if T has a model M in which P1(M) is κ1-like ordered, P2(M) is κ2-like ordered …, and Q1(M) is of power λ1, …, then T has a model N in which P1(M) is κ1′-like ordered …, Q1(N) is of power λ1′, …. (In this article κ is a strong-limit singular cardinal, and κ′ is a singular cardinal.)We also sometimes add the condition that M, N omits some types. The results are seemingly the best possible, i.e. according to our knowledge about n-cardinal problems (or, more precisely, a certain variant of them).

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 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 Universal graphs at the successor of a singular cardinal

2003 ◽  
Vol 68 (2) ◽  
pp. 366-388 ◽  
Author(s):  
Mirna Džamonja ◽  
Saharon Shelah
Keyword(s):  
General Problem ◽  
Supercompact Cardinal ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Consistency Results ◽  
Universal Graphs ◽  
General Method

AbstractThe paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are μ++ graphs on μ+ that taken jointly are universal for the graphs on μ+, while .The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.

scholarly journals CHAIN MODELS, TREES OF SINGULAR CARDINALITY AND DYNAMIC EF-GAMES

2011 ◽  
Vol 11 (01) ◽  
pp. 61-85 ◽  
Author(s):  
MIRNA DŽAMONJA ◽  
JOUKO VÄÄNÄNEN
Keyword(s):  
Order Logic ◽  
Chain Model ◽  
Descriptive Set Theory ◽  
Singular Cardinal ◽  
Infinitary Logic ◽  
Strong Limit ◽  
First Order ◽  
Winning Strategies ◽  
A Chain ◽  
Descriptive Set

Let κ be a singular cardinal. Karp's notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf (κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf (κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf (κ) = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf (κ)κ. The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models.

On skinny stationary subsets of

2013 ◽  
Vol 78 (2) ◽  
pp. 667-680 ◽  
Author(s):  
Yo Matsubara ◽  
Toshimichi Usuba
Keyword(s):  
Stationary Subset ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Stationary Subsets

AbstractWe introduce the notion of skinniness for subsets of and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2λ-saturation of NSκλ ∣ X, where NSκλ denotes the non-stationary ideal over , implies the existence of a skinny stationary subset of X. We also show that if λ is a singular cardinal, then there is no skinnier stationary subset of . Furthermore, if λ is a strong limit singular cardinal, there is no skinny stationary subset of . Combining these results, we show that if λ is a strong limit singular cardinal, then NSκλ ∣ X can satisfy neither precipitousness nor 2λ-saturation for every stationary X ⊆ . We also indicate that , where , is equivalent to the existence of a skinnier (or skinniest) stationary subset of under some cardinal arithmetical hypotheses.

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.

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