Knaster and friends II: The C-sequence number

Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the [Formula: see text]-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of [Formula: see text] and independence results about the [Formula: see text]-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general [Formula: see text]-sequence spectrum and uncover some tight connections between the [Formula: see text]-sequence spectrum and the strong coloring principle [Formula: see text], introduced in Part I of this series.

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On large cardinals and partition relations

Journal of Symbolic Logic ◽

10.2307/2270265 ◽

1971 ◽

Vol 36 (2) ◽

pp. 305-308 ◽

Cited By ~ 5

Author(s):

E. M. Kleinberg ◽

R. A. Shore

Keyword(s):

Set Theory ◽

Measurable Cardinal ◽

Order Type ◽

Large Cardinals ◽

Infinite Cardinal ◽

Partition Relation ◽

Weakly Compact ◽

Original Result ◽

Partition Relations ◽

Uncountable Cardinal

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.

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Tameness, powerful images, and large cardinals

Journal of Mathematical Logic ◽

10.1142/s0219061320500245 ◽

2020 ◽

Vol 21 (01) ◽

pp. 2050024

Author(s):

Will Boney ◽

Michael Lieberman

Keyword(s):

Large Cardinals ◽

Classification Theory ◽

Closure Properties ◽

Weakly Compact ◽

Accessible Categories ◽

Elementary Classes ◽

Comprehensive Level ◽

Theoretic Characterization ◽

Measurable Cardinals

We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc. 145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS 145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic 81(1) (2016) 1647–1648].

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On splitting stationary subsets of large cardinals

Journal of Symbolic Logic ◽

10.2307/2272121 ◽

1977 ◽

Vol 42 (2) ◽

pp. 203-214 ◽

Cited By ~ 25

Author(s):

James E. Baumgartner ◽

Alan D. Taylor ◽

Stanley Wagon

Keyword(s):

Large Cardinals ◽

Normal Ideal ◽

Weakly Compact ◽

Uncountable Cardinal ◽

Stationary Set ◽

Stationary Subsets ◽

Open Question ◽

Mahlo Cardinals ◽

Normal Ideals

AbstractLet κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ+-saturated, i.e., are there κ+ stationary subsets of κ with pairwise intersections nonstationary? Our first observation is:Theorem. NS isκ+-saturated iff for every normal ideal J on κ there is a stationary set A ⊆ κsuch that J = NS∣A = {X ⊆ κ: X ∩ A ∈ NS}.Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define “greatly Mahlo” cardinals and obtain the following:Theorem. If κ is greatly Mahlo then NS is notκ+-saturated.Theorem. If κ is ordinal Π11-indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries aκ-saturated ideal, thenκis greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π11-indescribable cardinal.These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ+-saturated.

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Synthesis and characterization of layered bismuth vanadates

Journal of Materials Research ◽

10.1557/jmr.1990.2718 ◽

1990 ◽

Vol 5 (11) ◽

pp. 2718-2722 ◽

Cited By ~ 56

Author(s):

K. B. R. Varma ◽

G. N. Subbanna ◽

T. N. Guru ◽

C. N. R. Rao

Keyword(s):

Curie Temperature ◽

General Formula ◽

Layered Structure ◽

Synthesis And Characterization ◽

Structure And Properties ◽

Bismuth Vanadates

Bi2VO5.5 (Bi4O11), which is the vanadium analog of the first member of the Aurivillius family of oxides of the general formula Bi2An−1BnO3n+3, has been prepared and characterized. The vanadate has the expected layered structure and is ferroelectric with a Curie temperature of 720 K. While we have not been able to synthesize the vanadium analog of the n = 2 member of the Aurivillius family, we have examined the structure and properties of a vanadate of the composition Bi2V3O9.

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Co-stationarity of the ground model

Journal of Symbolic Logic ◽

10.2178/jsl/1154698589 ◽

2006 ◽

Vol 71 (3) ◽

pp. 1029-1043 ◽

Cited By ~ 3

Author(s):

Natasha Dobrinen ◽

Sy-David Friedman

Keyword(s):

Large Cardinals ◽

Partial Ordering ◽

Proper Class ◽

Ground Model ◽

Covering Theorem ◽

Uncountable Cardinal ◽

Cohen Forcing ◽

Measurable Cardinals

AbstractThis paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

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Arene Ruthenium Metalla-Assemblies with Anthracene Moieties for PDT Applications

Inorganics ◽

10.3390/inorganics6030097 ◽

2018 ◽

Vol 6 (3) ◽

pp. 97 ◽

Cited By ~ 8

Author(s):

Marie Gaschard ◽

Farzaneh Nehzat ◽

Thomas Cheminel ◽

Bruno Therrien

Keyword(s):

Molecular Structure ◽

Single Crystal ◽

General Formula ◽

Structure Analysis ◽

White Light ◽

Light Irradiation ◽

Synthesis And Characterization ◽

X Ray

The synthesis and characterization of three metalla-rectangles of the general formula [Ru4(η6-p-cymene)4(μ4-clip)2(μ2-Lanthr)2][CF3SO3]4 (Lanthr: 9,10-bis(3,3’-ethynylpyridyl) anthracene; clip = oxa: oxalato; dobq: 2,5-dioxido-1,4-benzoquinonato; donq: 5,8-dioxido-1,4-naphthoquinonato) are presented. The molecular structure of the metalla-rectangle [Ru4(η6-p-cymene)4(μ4-oxa)2(μ2-Lanthr)2]4+ has been confirmed by the single-crystal X-ray structure analysis of [Ru4(η6-p-cymene)4(μ4-oxa)2(μ2-Lanthr)2][CF3SO3]4 · 4 acetone (A2 · 4 acetone), thus showing the anthracene moieties to be available for reaction with oxygen. While the formation of the endoperoxide form of Lanthr was observed in solution upon white light irradiation, the same reaction does not occur when Lanthr is part of the metalla-assemblies.

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Homogeneous changes in cofinalities with applications to HOD

Journal of Mathematical Logic ◽

10.1142/s0219061317500076 ◽

2017 ◽

Vol 17 (02) ◽

pp. 1750007 ◽

Cited By ~ 1

Author(s):

Omer Ben-Neria ◽

Spencer Unger

Keyword(s):

Large Cardinals ◽

Synthesis and characterization of new complexes of Co+2 , Ni+2 , Cu+2 and Zn+2 with (Sodium acetate thio ) ethylene ligands

Baghdad Science Journal ◽

10.21123/bsj.10.3.650-657 ◽

2013 ◽

Vol 10 (3) ◽

pp. 650-657

Author(s):

Baghdad Science Journal

Keyword(s):

Content Analysis ◽

General Formula ◽

Metal Content ◽

Sodium Acetate ◽

Magnetic Measurement ◽

Synthesis And Characterization ◽

Square Planar ◽

Square Planar Complexes ◽

Uv Visible

This research involves the preparation of new ligands 1,1,2,2- tetrakis (sodium acetate thio)ethylene(L1) and 1,1,2- tris(sodiumacetatethio) ethylene(L2), through the reaction of disodium thioglycolate) with tetra chloro ethylene or tri chloro ethylene in (1:4) or (1:3) moler ratio . Homodinucliar complexes of general formlu [M2(L1)] and [M2(L2)ClH2O] , when M= Co(II), Ni(II), Cu (II) and Zn(II) also mono nuclear complexes of general formula [M(L2)] . The prepared complexes were characterized using spectral method (UV/Visible/ IR) , metal content analysis , magnetic and atomic measurements . The spectral and magnetic measurement indicats that some complexes have tetrahedral or square planar complexes environtment .

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TAMENESS FROM LARGE CARDINAL AXIOMS

Journal of Symbolic Logic ◽

10.1017/jsl.2014.30 ◽

2014 ◽

Vol 79 (4) ◽

pp. 1092-1119 ◽

Cited By ~ 23

Author(s):

WILL BONEY

Keyword(s):

Large Cardinals ◽

Large Cardinal ◽

Strongly Compact Cardinal ◽

Weakly Compact ◽

Compact Cardinal ◽

Dual Property ◽

First Time ◽

Strongly Compact Cardinals ◽

Do So

AbstractWe show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC withLS(K) below a strongly compact cardinalκis <κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, calledtype shortness, and show that it follows similarly from large cardinals.

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Jensen's ⃞ principles and the Novák number of partially ordered sets

Journal of Symbolic Logic ◽

10.2307/2273941 ◽

1986 ◽

Vol 51 (1) ◽

pp. 47-58 ◽

Cited By ~ 13

Author(s):

Boban Veličković

Keyword(s):

Partial Order ◽

Partially Ordered Sets ◽

Large Cardinals ◽

Ordered Sets ◽

Weakly Compact ◽

Cardinal Function ◽

Partially Ordered ◽

Suslin Tree

In this paper we consider various properties of Jensen's □ principles and use them to construct several examples concerning the so-called Novák number of partially ordered sets.In §1 we give the relevant definitions and review some facts about □ principles. Apart from some simple observations most of the results in this section are known.In §2 we consider the Novák number of partially ordered sets and, using □ principles, give counterexamples to the productivity of this cardinal function. We also formulate a principle, show by forcing that it is consistent and use it to construct an ℵ2-Suslin tree T such that forcing with T × T collapses ℵ1.In §3 we briefly consider games played on partially ordered sets and relate them to the problems of the previous section. Using a version of □ we give an example of a proper partial order such that the game of length ω played on is undetermined.In §4 we raise the question of whether the Novák number of a homogenous partial order can be singular, and show that in some cases the answer is no.We assume familiarity with the basic techniques of forcing. In §1 some facts about large cardinals (e.g. weakly compact cardinals are -indescribable) and elementary properties of the constructible hierarchy are used. For this and all undefined terms we refer the reader to Jech [10].

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