Positive ∑ operations on ordinals and normal filters on greatly mahlo cardinals

1989 ◽  
Vol 54 (1) ◽  
pp. 226-233
Author(s):  
Thomas Jech
Keyword(s):  
Normal Filter ◽  
Uncountable Cardinal ◽  
Mahlo Cardinals ◽  
Ordinal Function

If ℱ is a normal filter on a regular uncountable cardinal κ, let ║f║ be the ℱ-norm of an ordinal function f. We introduce the class of positive ordinal operations and prove that if F is a positive operation then ║F(f)║ ≥ F(║f║). For each η < κ+ let fη be the canonical ηth function. We show that if F is a ∑ operation then F(fη) = fF(η).As an application we show that if κ is greatly Mahlo then there are normal filters on κ of order greater than κ+.

On splitting stationary subsets of large cardinals

1977 ◽  
Vol 42 (2) ◽  
pp. 203-214 ◽  
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.

Co-stationarity of the ground model

2006 ◽  
Vol 71 (3) ◽  
pp. 1029-1043 ◽  
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.

The spectrum of resplendency

1990 ◽  
Vol 55 (2) ◽  
pp. 626-636
Author(s):  
John T. Baldwin
Keyword(s):  
Homogeneous Model ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Recursive Theory ◽  
Finite Language

AbstractLet T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

The practical implementation of DIN EN ISO 16891 “Test methods for evaluating the degradation of characteristics of cleanable filter media“/Die praktische Umsetzung der Norm DIN EN ISO 16891 „Prüfmethode zur Ermittlung der Abnahme der Wirksamkeit von abreinigbaren Filtermedien“

Gefahrstoffe ◽  
2019 ◽  
Vol 79 (05) ◽  
pp. 176-180
Author(s):  
F. Schmidt ◽  
J. Weimann ◽  
C. König
Keyword(s):  
Test Methods ◽  
Practical Implementation ◽  
Filter Media ◽  
Uniform Loading ◽  
Normal Filter ◽  
Current Form ◽  
Safety Aspects ◽  
Test Conditions ◽  
Chemical Ageing ◽  
Praktische Umsetzung

Summary DIN EN ISO 16891:2016 “Test methods for evaluating the degradation of characteristics of cleanable filter media“ is the first standard in Germany that takes into account the thermal and chemical ageing of the filter media and stipulates how they are to be tested. These normative specifications were to be implemented as part of a research project. However, the boundary test conditions proved to be general conditions and many other details were not described in the standard. This is why, as well as there being many safety aspects, the filter testing has so far only been partially implemented. Uniform loading of several samples at the normal filter flow velocities used in practice could not be implemented. Doubt exists with regard to the comparability of the results of the tests that were based on the standard in its current form at different test institutes.

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.

On the role of Ramsey quantifiers in first order arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 423-435 ◽  
Author(s):  
James H. Schmerl ◽  
Stephen G. Simpson
Keyword(s):  
Formal System ◽  
Intended Meaning ◽  
Completeness Proof ◽  
First Order ◽  
Uncountable Cardinal ◽  
Consistent Extension ◽  
Witness Set ◽  
Order Arithmetic ◽  
Infinite Set

The purpose of this paper is to study a formal system PA(Q2) of first order Peano arithmetic, PA, augmented by a Ramsey quantifier Q2 which binds two free variables. The intended meaning of Q2xx′φ(x, x′) is that there exists an infinite set X of natural numbers such that φ(a, a′) holds for all a, a′ Є X such that a ≠ a′. Such an X is called a witness set for Q2xx′φ(x, x′). Our results would not be affected by the addition of further Ramsey quantifiers Q3, Q4, …, Here of course the intended meaning of Qkx1 … xkφ(x1,…xk) is that there exists an infinite set X such that φ(a1…, ak) holds for all k-element subsets {a1, … ak} of X.Ramsey quantifiers were first introduced in a general model theoretic setting by Magidor and Malitz [13]. The system PA{Q2), or rather, a system essentially equivalent to it, was first defined and studied by Macintyre [12]. Some of Macintyre's results were obtained independently by Morgenstern [15]. The present paper is essentially self-contained, but all of our results have been directly inspired by those of Macintyre [12].After some preliminaries in §1, we begin in §2 by giving a new completeness proof for PA(Q2). A by-product of our proof is that for every regular uncountable cardinal k, every consistent extension of PA(Q2) has a k-like model in which all classes are definable. (By a class we mean a subset of the universe of the model, every initial segment of which is finite in the sense of the model.)

Killing them softly: degrees of inaccessible and Mahlo cardinals

2017 ◽  
Vol 63 (3-4) ◽  
pp. 256-264
Author(s):  
Erin Kathryn Carmody
Keyword(s):  
Mahlo Cardinals

scholarly journals Consequences of Choosing Different Settings When Processing Hip-Based Accelerometry Data From Older Adults: A Practical Approach Using Baseline Data From the SITLESS Study

2020 ◽  
Vol 3 (2) ◽  
pp. 89-99 ◽  
Author(s):  
Jason J. Wilson ◽  
Mathias Skjødt ◽  
Ilona McMullan ◽  
Nicole E. Blackburn ◽  
Maria Giné-Garriga ◽  
...  
Keyword(s):  
Older Adults ◽  
Repeated Measures ◽  
Low Frequency ◽  
Vertical Axis ◽  
Time Algorithm ◽  
Normal Filter ◽  
Wear Time ◽  
Epoch Length ◽  
Vector Magnitude ◽  
Selection Of

Accurately measuring older adults’ physical activity (PA) and sedentary behavior (SB) using accelerometers is essential, as both are important markers of health. This study aimed to highlight how steps taken during data processing may affect key hip-based accelerometry outcomes in older adults, using a selection of baseline accelerometry data (n = 658) from the SITLESS study. Different analytical parameters tested included wear-time algorithms, use of low-frequency extension (LFE) filter, epoch length, and minimum and maximum daily wear-time thresholds. These were compared against vertical axis counts per minute (CPM), vector magnitude (VM) CPM, SB, light PA, moderate-to-vigorous PA, step counts, and wear-time percentage. Differences in settings across the analytical parameters were assessed using paired sample t-tests and repeated measures ANOVAs using Bonferroni correction. Using the “Choi” versus “Troiano” wear-time algorithm resulted in a higher percentage wear-time. Most SB and PA outcomes were significantly different across wear-time algorithms (p < .001). This was similar when using the LFE filter versus normal filter (p < .001). Using 10-second epoch length increased daily SB time (between +75.7 and +79.2 minutes) compared to 60-second. Most SB and PA outcomes significantly changed comparing minimum-wear-time thresholds of 360, 480, 600, and 720 minutes per day (p < .001). Applying a log-diary with a ≥1140-minute threshold had a significant impact on vertical axis CPM, VM CPM, SB, and light PA outcomes (p < .001). This study demonstrates the potential variability in the number of participants being included in studies and reported SB and PA levels when processing older adults’ accelerometry data dependent on the analytical procedures utilized.

scholarly journals Distributive and related ideals in generic extensions

1987 ◽  
Vol 106 ◽  
pp. 91-100
Author(s):  
C. A. Johnson
Keyword(s):  
Quotient Algebra ◽  
Complete Ideal ◽  
Naturally Occurring ◽  
Uncountable Cardinal ◽  
Game Theoretic ◽  
Generic Extensions

Let κ: be a regular uncountable cardinal and I a κ-complete ideal on te. In [11] Kanai proved that the μ-distributivity of the quotient algebra P(κ)I is preserved under κ-C.C. μ-closed forcing. In this paper we extend Kanai’s result and also prove similar preservation results for other naturally occurring forms of distributivity. We also consider the preservation of two game theoretic properties of I and in particular, using a game theoretic equivalent of precipitousness we give a new proof of Kakuda’s theorem ([10]) that the precipitousness of I is preserved under κ-C.C. forcing.

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