Partition properties and Prikry forcing on simple spaces

1990 ◽  
Vol 55 (3) ◽  
pp. 938-947
Author(s):  
J. M. Henle
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Short Introduction ◽  
Prikry Forcing ◽  
Partition Property ◽  
Combinatorial Objects ◽  
The Relationship

One of the simplest and yet most fruitful ideas in forcing was the notion of Karel Prikry in which he used a measure on a cardinal κ to change the cofinality of κ to ω without collapsing it. The idea has found connections to almost every branch of modern set theory, from large cardinals to small, from combinatorics to models, from Choice to Determinacy, and from consistency to inconsistency. The long list of generalizers and modifiers includes Apter, Gitik, Henle, Spector, Shelah, Mathias, Magidor, Radin, Blass and Kimchi.This paper is about generalizing Prikry forcing and partition properties to “simple spaces”. The concept of a simple space is itself the generalization of those combinatorial objects upon which the notions of “measurable”, “compact”, “supercompact”, “huge”, etc. are based. Simple spaces were introduced in [ADHZ1] and [ADHZ2] together with a broader generalization, “filter spaces”. The definition provided here is a small simplification of earlier versions. The author is indebted to Mitchell Spector, whose careful reading turned up numerous errors, some subtle, some flagrant.In this first section, we review simple spaces briefly, including a short introduction to the space Qκλ. In §2, we describe our generalizations of partition property and Prikry forcing, and discuss the relationship between them. In §3, we find a partition property for the huge space [λ]κ, but show that Prikry forcing here is impossible. We find partition properties for Qκλ and show that Prikry forcing can be done here.

scholarly journals Internal Consistency and the Inner Model Hypothesis

2006 ◽  
Vol 12 (4) ◽  
pp. 591-600 ◽  
Author(s):  
Sy-David Friedman
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Weak Sense ◽  
Large Cardinals ◽  
Model Method ◽  
Inner Models ◽  
Inner Model ◽  
The Relationship ◽  
The Universe ◽  
The Way

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening of Cohen's result:Theorem 1 (Easton's Theorem). There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals.

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Los Angeles ◽  
Set Theory ◽  
Model Theory ◽  
Large Cardinals ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Definable Sets ◽  
Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

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 Postfeminism, popular feminism and neoliberal feminism? Sarah Banet-Weiser, Rosalind Gill and Catherine Rottenberg in conversation

2019 ◽  
Vol 21 (1) ◽  
pp. 3-24 ◽  
Author(s):  
Sarah Banet-Weiser ◽  
Rosalind Gill ◽  
Catherine Rottenberg
Keyword(s):  
Short Introduction ◽  
Popular Feminism ◽  
The Relationship

In this unconventional article, Sarah Banet-Weiser, Rosalind Gill and Catherine Rottenberg conduct a three-way ‘conversation’ in which they all take turns outlining how they understand the relationship among postfeminism, popular feminism and neoliberal feminism. It begins with a short introduction, and then Ros, Sarah and Catherine each define the term they have become associated with. This is followed by another round in which they discuss the overlaps, similarities and disjunctures among the terms, and the article ends with how each one understands the current mediated feminist landscape.

scholarly journals Chains of end elementary extensions of models of set theory

1998 ◽  
Vol 63 (3) ◽  
pp. 1116-1136 ◽  
Author(s):  
Andrés Villaveces
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Elementary Extension ◽  
Models Of Set Theory

AbstractLarge cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (‘unfoldable cardinals’) lie in the boundary of the propositions consistent with ‘V = L’ and the existence of 0#. We also provide an ‘embedding characterisation’ of the unfoldable cardinals and study their preservation and destruction by various forcing constructions.

Anthropocene: A Very Short Introduction

2018 ◽  
Author(s):  
Erle C. Ellis
Keyword(s):  
Climate Change ◽  
Mass Extinction ◽  
Scientific Community ◽  
Geological Time ◽  
Short Introduction ◽  
Geological Record ◽  
Species Invasions ◽  
Rapid Climate Change ◽  
The Relationship

Humanity’s impact on the planet has been profound. From fire, intensive hunting, and agriculture, it has accelerated into rapid climate change, widespread pollution, plastic accumulation, species invasions, and the mass extinction of species—changes that have left a permanent mark in the geological record of the rocks. Yet the proposal for a new unit of geological time—the Anthropocene Epoch—has raised debate far beyond the scientific community. The Anthropocene has emerged as a powerful new narrative of the relationship between humans and nature. Anthropocene: A Very Short Introduction draws on the work of geologists, geographers, environmental scientists, archaeologists, and humanities scholars to explain the science and wider implications of the Anthropocene.

scholarly journals Analysis of the acoustic climate of a spa park using the fuzzy set theory

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
Małgorzata Sztubecka ◽  
Jacek Sztubecki
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Urban Areas ◽  
Environmental Pollutants ◽  
Sound Level ◽  
The Road ◽  
The People ◽  
The Individual ◽  
The Relationship

Abstract The paper describes the differences between the actual results of the measurement of equivalent sound level and the feelings of people visiting "a Spa Park". Noise, as one of the environmental pollutants, cause detrimental effects on the recipient. Measurements of noise are usually performed in urban areas, especially in the road environments, providing a basis for measures to limit their impact on the environment. Often in the measurement there are ignored areas for recreation. Usually, they do not determine the relationship between the results of measurements of noise equivalent sound level and the individual feelings of the people living in these areas. The analysis was performed with the use of fuzzy set theory. The evaluation of the acoustic climate on the "Spa Park" should be determined on the basis of sound level measurements and questionnaires.

A combinatorial property of pκλ

1976 ◽  
Vol 41 (1) ◽  
pp. 225-234
Author(s):  
Telis K. Menas
Keyword(s):  
Large Cardinals ◽  
Combinatorial Property ◽  
Combinatorial Properties ◽  
Partition Property ◽  
Homogeneous Set ◽  
Unbounded Subset

In a paper on combinatorial properties and large cardinals [2], Jech extended several combinatorial properties of a cardinal κ to analogous properties of the set of all subsets of λ of cardinality less than κ, denoted by “pκλ”, where λ is any cardinal ≤κ. We shall consider in this paper one of these properties which is historically rooted in a theorem of Ramsey [10] and in work of Rowbottom [12].As in [2], define [pκλ]2 = {{x, y}: x, y ∈ pκλ and x ≠ y}. An unbounded subset A of pκλ is homogeneous for a function F: [pκλ]2 → 2 if there is a k < 2 so that for all x, y ∈ A with either x ⊊ y or y ⊊ x, F({x, y}) = k. A two-valued measure ü on pκλ is fine if it is κ-complete and if for all α < λ, ü({x ∈ pκλ: α ∈ x}) = 1, and ü is normal if, in addition, for every function f: pκλ → λsuch that ü({x ∈ pκλ: f(x) ∈ x}) = 1, there is an α < λ such that ü({x ∈ pκλ: f(x) = α}) = 1. Finally, a fine measure on pκλ has the partition property if every F: [pκλ]2 → 2 has a homogeneous set of measure one.

scholarly journals IDEAL PROJECTIONS AND FORCING PROJECTIONS

2014 ◽  
Vol 79 (4) ◽  
pp. 1247-1285 ◽  
Author(s):  
SEAN COX ◽  
MARTIN ZEMAN
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Normal Ideal ◽  
List Type ◽  
Type Number ◽  
Inner Model ◽  
Image Position ◽  
Open Question ◽  
Woodin Cardinal

AbstractIt is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:(1)If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;(2)For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

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