scholarly journals On the partition property of measures onPℋλ

1982 ◽  
Vol 5 (4) ◽  
pp. 817-821
Author(s):  
Donald H. Pelletier
Keyword(s):  
Measurable Cardinal ◽  
Combinatorial Property ◽  
Partition Property ◽  
Normal Measure

The partition property for measures onPℋλwas formulated by analogy with a property which Rowbottom [1] proved was possessed by every normal measure on a measurable cardinal. This property has been studied in [2], [3], and [4]. This note summarizes [5] and [6], which contain results relating the partition property with the extendibility of the measure and with an auxiliary combinatorial property introduced by Menas in [4]. Detailed proofs will appear in [5] and [6].

On the ultrafilters and ultrapowers of strong partition cardinals

1984 ◽  
Vol 49 (4) ◽  
pp. 1268-1272
Author(s):  
J.M. Henle ◽  
E.M. Kleinberg ◽  
R.J. Watro
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Partition Property ◽  
Axiom Of Determinacy ◽  
Almost Everywhere ◽  
Normal Measure ◽  
Constructible Sets ◽  
Inner Model ◽  
The Axiom Of Choice ◽  
Strong Partition Cardinals

A strong partition cardinal is an uncountable well-ordered cardinal κ such that every partition of [κ]κ (the size κ subsets of κ) into less than κ many pieces has a homogeneous set of size κ. The existence of such cardinals is inconsistent with the axiom of choice, and our work concerning them is carried out in ZF set theory with just dependent choice (DC). The consistency of strong partition cardinals with this weaker theory remains an open question. The axiom of determinacy (AD) implies that a large number of cardinals including ℵ1 have the strong partition property. The hypothesis that AD holds in the inner model of constructible sets built over the real numbers as urelements has important consequences for descriptive set theory, and results concerning strong partition cardinals are often applied in this context. Kechris [4] and Kechris et al. [5] contain further information concerning the relationship between AD and strong partition cardinals.We assume familiarity with the basic results on strong partition cardinals as developed in Kleinberg [6], [7], [8] and Henle [2]. Recall that a strong partition cardinal κ is measurable; in fact every stationary subset of κ is measure one under some normal measure on κ. If μ is a countably additive ultrafilter extending the closed unbounded filter on κ, then the length of the ultrapower [κ]κ under the less than almost everywhere μ ordering is again a measurable cardinal. In §1 below we establish a polarized partition property on these measurable cardinals.

A Cardinal Structure Theorem for an Ultrapower

1985 ◽  
Vol 28 (4) ◽  
pp. 472-473
Author(s):  
Arthur W. Apter
Keyword(s):  
Structure Theorem ◽  
Measurable Cardinal ◽  
Normal Measure

AbstractIn this note, we construct a model with a normal measure U over a measurable cardinal κ so that the cardinal structures of V and Vκ/U are the same ≤2κ. We then show that it is possible to construct a model where this is not true.

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.

Coding over a measurable cardinal

1989 ◽  
Vol 54 (4) ◽  
pp. 1145-1159
Author(s):  
Sy D. Friedman
Keyword(s):  
Initial Segment ◽  
Measurable Cardinal ◽  
Large Cardinals ◽  
Main Difficulty ◽  
Normal Measure ◽  
Coding Method ◽  
Almost Disjoint

The purpose of this paper is to extend the coding method (see Beller, Jensen and Welch [82]) into the context of large cardinals.Theorem. Suppose μ is a normal measure on κ in V and 〈 V, A〉 ⊨ ZFC. Then there is a 〈V, A〉-definable forcing for producing a real R such that:(a) V[R] ⊨ ZFC and A is V[R]-definable with parameter R.(b) V[R] = L[μ*, R], where μ* is a normal measure on κ in V[R] extending μ.(c) V ⊨ GCH → is cardinal and cofinality preserving.Corollary. It is consistent that μ is a normal measure, R ⊆ ω is not set-generic over L[μ] and 0+ ∉ L[μ, R].Some other corollaries will be discussed in §4 of the paper.The main difficulty in L[μ]-coding lies in the problem of “stationary restraint”.As in all coding constructions, conditions will be of the form belonging to an initial segment of the cardinals, where p(γ) is a condition for almost disjoint coding into a subset of γ+. In addition for limit cardinals γ in Domain(p), 〈pγ′∣γ′ < γ〉 serves to code pγ.An important restriction in coding arguments is that for inaccessible for only a nonstationary set of γ′ < γ. The reason is that otherwise there are conflicts between the restraint imposed by the different and the need to code extensions of pγ below γ.

On a combinatorial property of menas related to the partition property for measures on supercompact cardinals

1983 ◽  
Vol 48 (2) ◽  
pp. 475-481 ◽  
Author(s):  
Kenneth Kunen ◽  
Donald H. Pelletier
Keyword(s):  
Supercompact Cardinal ◽  
Combinatorial Property ◽  
Supercompact Cardinals ◽  
Partition Property

AbstractT.K. Menas [4, pp. 225–234] introduced a combinatorial property Χ(μ) of a measure μ on a supercompact cardinal κ and proved that measures with this property also have the partition property. We prove here that Menas' property is not equivalent to the partition property. We also show that if a is the least cardinal greater than κ such that Pκα bears a measure without the partition property, then α is inaccessible and -indescribable.

Ramsey cardinals, α-Erdös cardinals, and the core model

1991 ◽  
Vol 56 (1) ◽  
pp. 108-114
Author(s):  
Dirk R. H. Schlingmann
Keyword(s):  
Measurable Cardinal ◽  
Core Model ◽  
The Core ◽  
Transitive Model ◽  
Normal Measure ◽  
Inner Model

The core model K was introduced by R. B. Jensen and A. J. Dodd [DoJ]. K is the union of Gödel's constructible universe L together with all mice, i.e., , and K is a transitive model of ZFC + (V = K) + GCH (see [DoJ]). V = K is consistent with the existence of Ramsey cardinals [M], and if cf(α) > ω, V = K is consistent with the existence of α-Erdös cardinals [J]. Let K be Ramsey. Then there is a smallest inner model Wκ of ZFC in which κ is Ramsey. We have Wκ ⊨ V = K and Wκ ⊆ K [M]. The existence of Wκ with is equivalent to the existence of a sharplike mouse on N ⊨ K with N ⊨ κ Ramsey. (A mouse N on is called sharplike provided .) We have , where is the mouse iteration of N. N is the oleast mouse not in Wκ (see [J] and [DJKo]). Here < denotes the mouse order. The context always clarifies whether the mouse order or the usual <-relation is meant.The main result of §1 is that Wκ ⊨ κ is the only Ramsey cardinal. A similar result has been found true in the smallest inner model L[U] of ZFC + “κ is measurable” if U is a normal measure on κ: L[U] ⊨ κ is the only measurable cardinal [Ku].

On the relationship between the partition property and the weak partition property for normal ultrafilters on Pκλ

1993 ◽  
Vol 58 (1) ◽  
pp. 119-127
Author(s):  
Julius B. Barbanel
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
The Other ◽  
Supercompact Cardinal ◽  
Partition Property ◽  
Other Hand ◽  
The One ◽  
Measurable Cardinals ◽  
The Relationship ◽  
Normal Ultrafilter

AbstractSuppose κ is a supercompact cardinal and λ > κ. We study the relationship between the partition properly and the weak partition properly for normal ultrafilters on Pκλ. On the one hand, we show that the following statement is consistent, given an appropriate large cardinal assumption: The partition property and the weak partition properly are equivalent, there are many normal ultrafilters that satisfy these properties, and there are many normal ultrafilters that do not satisfy these properties. On the other hand, we consider the assumption that, for some λ > κ, there exists a normal ultrafilter U on Pκλ such that U satisfies the weak partition property but does not satisfy the partition property. We show that this assumption is implied by the assertion that there exists a cardinal γ > κ such that γ is γ+-supercompact, and, assuming the GCH, it implies the assertion that there exists a cardinal γ > κ such that γ is a measurable cardinal with a normal ultrafilter concentrating on measurable cardinals.

The ⊲-ordering on normal ultrafilters

1985 ◽  
Vol 50 (4) ◽  
pp. 936-952 ◽  
Author(s):  
Stewart Baldwin
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
The Other ◽  
Ground Model ◽  
Other Hand ◽  
Normal Measure ◽  
Inner Model ◽  
Normal Ultrafilter

If κ is a measurable cardinal, then it is a well-known fact that there is at least one normal ultrafilter over κ. In [K-1], Kunen showed that one cannot say more without further assumptions, for if U is a normal ultrafilter over κ, then L[U] is an inner model of ZFC in which κ has exactly one normal measure. On the other hand, Kunen and Paris showed [K-P] that if κ is measurable in the ground model, then there is a forcing extension in which κ has normal ultrafilters, so it is consistent that κ has the maximum possible number of normal ultrafilters. Starting with assumptions stronger than measurability, Mitchell [Mi-1] filled in the gap by constructing models of ZFC + GCH satisfying “there are exactly λ normal ultrafilters over κ”, where λ could be κ+ or κ++ (measured in the model), or anything ≤ κ. Whether or not Mitchell's results can be obtained by starting only with a measurable cardinal in the ground model and defining a forcing extension is unknown.There are substantial differences between the Mitchell models and the Kunen-Paris models. In the Kunen-Paris models κ can be the only measurable cardinal. However, in the Mitchell model in which κ has exactly 2 normal ultrafilters, one of them contains the set {α < κ: α is measurable} while the other does not. Thus it is natural to ask if it is possible to get a model M of ZFC in which κ is the only measurable cardinal and κ has exactly 2 normal ultrafilters. In this paper we will show that, using appropriate large cardinal assumptions, the answer is yes.

Inflammatory cytokine inhibition of erythropoiesis in patients implanted with a mechanical circulatory assist device

Perfusion ◽  
2005 ◽  
Vol 20 (2) ◽  
pp. 83-90 ◽  
Author(s):  
Christopher N Pierce ◽  
Douglas F Larson
Keyword(s):  
Inflammatory Cytokine ◽  
Hematological Parameters ◽  
Hemoglobin Concentration ◽  
Mean Cell Volume ◽  
Distribution Width ◽  
Normal Limits ◽  
Circulatory Assist Devices ◽  
Cytokine Inhibition ◽  
Normal Measure ◽  
The University

Mechanical circulatory assist devices (MCADs) are increasingly utilized independently of cardiac transplantation in the management of heart failure. Though MCAD use incorporates inherent mechanical risks, the inevitable onset of chronic anemia, with its associated morbidity and mortality, is also a significant concern. MCAD support has been correlated with elevated plasma levels of inflammatory cytokines TNF-α, IL-1b, and IL-6, which have separately been found to inhibit erythropoietin (Epo)-induced erythrocyte (RBC) maturation. Previous analysis of hematological parameters for MCAD-sup-ported patients concluded that an amplified inflammatory response impedes RBC proliferation and recovery from hemolytic anemia. Additional analysis may bolster this assertion. Hemoglobin concentration (HC), RBC distribution width (RDW), mean cell volume (MCV), and cardiac index were retrospectively analysed for 78 MCAD-supported patients implanted for greater than 30 days at the University of Arizona Health Sciences Center from 1996 to 2002. Analysis confirms that the HC, a conventional marker for anemia, declines with MCAD placement and remains below the clinically defined, minimum normal value. Inversely, the RDW rises above maximum normal measure, signifying an increased fraction of juvenile RBCs. The MCV remains unchanged and within normal limits, demonstrating adequate substrate for RBC formation. MCAD performance also stabilizes as adequate perfusion returns. These results further support our previously published conclusion that a sufficient response of erythropoiesis occurs in reaction to the onset of anemia by an increased production of immature RBCs. However, the cells never fully mature and join circulation. The patient’s inflammatory cytokine response to the implanted device most likely mediates the chronic MCAD-induced anemia by inhibition of Epo effects.

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