Ultrafilters generated by a closed set of functions

1995 ◽  
Vol 60 (2) ◽  
pp. 415-430
Author(s):  
Greg Bishop
Keyword(s):  
Regular Cardinal ◽  
The Other ◽  
Strong Limit ◽  
Closed Set ◽  
Limit Cardinal ◽  
Regular Cardinals ◽  
Increasing Functions

AbstractLet κ and λ be infinite cardinals, a filter on κ and a set of functions from κ to κ. The filter is generated by if consists of those subsets of κ which contain the range of some element of . The set is <λ-closed if it is closed in the <λ-topology on κκ. (In general, the <λ-topology on IA has basic open sets all such that, for all i ∈ I, Ui ⊆ A and ∣{i ∈ I: Ui ≠ A} ∣<λ.) The primary question considered in this paper asks “Is there a uniform ultrafilter on κ which is generated by a closed set of functions?” (Closed means <ω-closed.) We also establish the independence of two related questions. One is due to Carlson: “Does there exist a regular cardinal κ and a subtree T of <κκ such that the set of branches of T generates a uniform ultrafilter on κ?”; and the other is due to Pouzet: “For all regular cardinals κ, is it true that no uniform ultrafilter on κ is it true that no uniform ultrafilter on κ analytic?”We show that if κ is a singular, strong limit cardinal, then there is a uniform ultrafilter on κ which is generated by a closed set of increasing functions. Also, from the consistency of an almost huge cardinal, we get the consistency of CH + “There is a uniform ultrafilter on ℵ1 which is generated by a closed set of increasing functions”. In contrast with the above results, we show that if Κ is any cardinal, λ is a regular cardinal less than or equal to κ and ℙ is the forcing notion for adding at least (κ<λ)+ generic subsets of λ, then in VP, no uniform ultrafilter on κ is generated by a closed set of functions.

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.

The tree property at ℵω+2

2011 ◽  
Vol 76 (2) ◽  
pp. 477-490 ◽  
Author(s):  
Sy-David Friedman ◽  
Ajdin Halilović
Keyword(s):  
Strong Limit ◽  
Tree Property ◽  
Weakly Compact ◽  
Limit Cardinal

AbstractAssuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension ℵω is a strong limit cardinal and ℵω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).

scholarly journals Recognition of Synthetic Speech by Hearing-Impaired Elderly Listeners

1991 ◽  
Vol 34 (5) ◽  
pp. 1180-1184 ◽  
Author(s):  
Larry E. Humes ◽  
Kathleen J. Nelson ◽  
David B. Pisoni
Keyword(s):  
Hearing Loss ◽  
Speech Recognition ◽  
Recognition Performance ◽  
The Elderly ◽  
Hearing Impaired ◽  
The Other ◽  
Natural Speech ◽  
Synthetic Speech ◽  
Closed Set ◽  
Open Set

The Modified Rhyme Test (MRT), recorded using natural speech and two forms of synthetic speech, DECtalk and Votrax, was used to measure both open-set and closed-set speech-recognition performance. Performance of hearing-impaired elderly listeners was compared to two groups of young normal-hearing adults, one listening in quiet, and the other listening in a background of spectrally shaped noise designed to simulate the peripheral hearing loss of the elderly. Votrax synthetic speech yielded significant decrements in speech recognition compared to either natural or DECtalk synthetic speech for all three subject groups. There were no differences in performance between natural speech and DECtalk speech for the elderly hearing-impaired listeners or the young listeners with simulated hearing loss. The normal-hearing young adults listening in quiet out-performed both of the other groups, but there were no differences in performance between the young listeners with simulated hearing loss and the elderly hearing-impaired listeners. When the closed-set identification of synthetic speech was compared to its open-set recognition, the hearing-impaired elderly gained as much from the reduction in stimulus/response uncertainty as the two younger groups. Finally, among the elderly hearing-impaired listeners, speech-recognition performance was correlated negatively with hearing sensitivity, but scores were correlated positively among the different talker conditions. Those listeners with the greatest hearing loss had the most difficulty understanding speech and those having the most trouble understanding natural speech also had the greatest difficulty with synthetic speech.

NUMERICAL SIMULATION ON THE EXISTENCE OF FLUCTUATION OF INSTANTANEOUS AVAILABILITY

2016 ◽  
Vol 40 (5) ◽  
pp. 703-713 ◽  
Author(s):  
Yang Yi ◽  
Ren Si Chao ◽  
Fan Guimei ◽  
Kang Rui
Keyword(s):  
Numerical Simulation ◽  
Numerical Methods ◽  
Failure Rate ◽  
The Other ◽  
Repairable System ◽  
Repair Rate ◽  
Trapezoidal Formula ◽  
Other Hand ◽  
Increasing Functions

This paper considers the fluctuation of the instantaneous availability by numerical methods for a one-unit repairable system. The choices of the failure rate and repair rate are linear or cubic increasing functions. For the equation of instantaneous availability composing of two convolutions, the following numerical methods are used: the composite Simpson formula and the trapezoidal formula. That is to say, the simulated curves of instantaneous availability under any condition are obtained. Through the simulated results, when the failure rate and repair rate are selected as increasing functions, the extremum of simulated curve exists so fluctuation exists. On the other hand, if parameters of increasing functions become smaller, the fluctuation weakens.

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (2) ◽  
pp. 419-426
Author(s):  
Manuel Lerman
Keyword(s):  
Lattice Theory ◽  
Regular Cardinal ◽  
Finite Sets ◽  
First Order ◽  
Recursively Enumerable ◽  
Regular Cardinals ◽  
Constructible Sets ◽  
Set Inclusion ◽  
Order Language ◽  
Recursively Enumerable Sets

Let α be an admissible ordinal, and let (α) denote the lattice of α-r.e. sets, ordered by set inclusion. An α-r.e. set A is α*-finite if it is α-finite and has ordertype less than α* (the Σ1 projectum of α). An a-r.e. set S is simple if (the complement of S) is not α*-finite, but all the α-r.e. subsets of are α*-finite. Fixing a first-order language ℒ suitable for lattice theory (see [2, §1] for such a language), and noting that the α*-finite sets are exactly those elements of (α), all of whose α-r.e. subsets have complements in (α) (see [4, p. 356]), we see that the class of simple α-r.e. sets is definable in ℒ over (α). In [4, §6, (Q22)], we asked whether an admissible ordinal α exists for which all simple α-r.e. sets have the same 1-type. We were particularly interested in this question for α = ℵ1L (L is Gödel's universe of constructible sets). We will show that for all α which are regular cardinals of L (ℵ1L is, of course, such an α), there are simple α-r.e. sets with different 1-types.The sentence exhibited which differentiates between simple α-r.e. sets is not the first one which comes to mind. Using α = ω for intuition, one would expect any of the sentences “S is a maximal α-r.e. set”, “S is an r-maximal α-r.e. set”, or “S is a hyperhypersimple α-r.e. set” to differentiate between simple α-r.e. sets. However, if α > ω is a regular cardinal of L, there are no maximal, r-maximal, or hyperhypersimple α-r.e. sets (see [4, Theorem 4.11], [5, Theorem 5.1] and [1,Theorem 5.21] respectively). But another theorem of (ω) points the way.

Weakly normal closures of filters on Pκλ

1993 ◽  
Vol 58 (1) ◽  
pp. 55-63 ◽  
Author(s):  
Masahiro Shioya
Keyword(s):  
Regular Cardinal ◽  
Natural Generalization ◽  
Large Cardinal ◽  
The Other ◽  
Disjoint Family ◽  
Straightforward Generalization ◽  
Weak Normality ◽  
Natural Formulation

The study of filters on Pκλ started by Jech [5] as a natural generalization of that of filters on an uncountable regular cardinal κ. Several notions including weak normality have been generalized. However, there are two versions proposed as weak normality for filters on Pκλ. One is due to Abe [1] as a straightforward generalization of weak normality for filters on κ due to Kanamori [6] and the other is due to Mignone [8]. While Mignone's version is weaker than normality, Kanamori-Abe's version is not in general. In fact, Abe [2] has proved, generalizing Kanamori [6], that a filter is weakly normal in the sense of Abe iff it is weakly normal in the sense of Mignone and there exists no disjoint family of cfλ-many positive sets. Therefore Kanamori-Abe's version is essentially a large cardinal property and Mignone's version seems to be the most natural formulation of “weak” normality.In this paper, we study weak normality in the sense of Mignone. In [8], Mignone studies weak normality of canonically defined filters. We complement his chart and try to find the weakly normal closures of these filters (i.e., the minimal weakly normal filters extending them). Therefore our result is a natural refinement of Carr [4].It is now well known that combinatorics on Pκλ is not a naive generalization of that on κ. For example, Menas [7] showed that stationarity on Pκλ can be characterized by 2-dimensional regressive functions, but not by 1-dimensional ones when λ is strictly larger than κ. We show in terms of weak normality that combinatorics on Pκλ vary drastically with respect to cfλ.

Minimal α-recursion theoretic degrees

1973 ◽  
Vol 38 (1) ◽  
pp. 18-28 ◽  
Author(s):  
John M. MacIntyre
Keyword(s):  
Recursive Function ◽  
Regular Cardinal ◽  
Recursion Theory ◽  
Minimal Degree ◽  
The Other ◽  
Ordinal Numbers ◽  
Definition Of

This paper investigates the problem of extending the recursion theoretic construction of a minimal degree to the Kripke [2]-Platek [5] recursion theory on the ordinals less than an admissible ordinal α, a theory derived from the Takeuti [11] notion of a recursive function on the ordinal numbers. As noted in Sacks [7] when one generalizes the recursion theoretic definition of relative recursiveness to α-recursion theory for α > ω the two usual definitions give rise to two different notions of reducibility. We will show that whenever α is either a countable admissible or a regular cardinal of the constructible universe there is a subset of α whose degree is minimal for both notions of reducibility. The result is an excellent example of a theorem of ordinary recursion theory obtainable via two different constructions, one of which generalizes, the other of which does not. The construction which cannot be lifted to α-recursion theory is that of Spector [10]. We sketch the reasons for this in §3.

scholarly journals Large semigroups of cellular automata

2011 ◽  
Vol 32 (6) ◽  
pp. 1991-2010 ◽  
Author(s):  
YAIR HARTMAN
Keyword(s):  
Cellular Automata ◽  
Field Structure ◽  
The Other ◽  
Entire Space ◽  
Dimensional Torus ◽  
Semigroup Action ◽  
Closed Set ◽  
One Dimensional ◽  
Shift Space ◽  
The One

AbstractIn this article, we consider semigroups of transformations of cellular automata which act on a fixed shift space. In particular, we are interested in two properties of these semigroups which relate to ‘largeness’: first, a semigroup has the ID (infinite is dense) property if the only infinite invariant closed set (with respect to the semigroup action) is the entire space; the second property is maximal commutativity (MC). We shall consider two examples of semigroups: one is spanned by cellular automata transformations that represent multiplications by integers on the one-dimensional torus, and the other one consists of all the cellular automata transformations which are linear (when the symbols set is the ring ℤ/sℤ). It will be shown that these two properties of these semigroups depend on the number of symbols s. The multiplication semigroup is ID and MC if and only if s is not a power of a prime. The linear semigroup over the mentioned ring is always MC but is ID if and only if s is prime. When the symbol set is endowed with a finite field structure (when possible), the linear semigroup is both ID and MC. In addition, we associate with each semigroup which acts on a one-sided shift space a semigroup acting on a two-sided shift space, and vice versa, in a way that preserves the ID and the MC properties.

A note on cardinal exponentiation

1980 ◽  
Vol 45 (1) ◽  
pp. 56-66 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Strong Limit ◽  
Limit Cardinal

AbstractSilver and subsequently Galvin and Hajnal, got bounds on , for ℵα strong limit cardinal of cofinality > ℵ0. We somewhat improve those results.

Wild edge colourings of graphs

2004 ◽  
Vol 69 (1) ◽  
pp. 255-264
Author(s):  
Mirna Džamonja ◽  
Péter Komjáth ◽  
Charles Morgan
Keyword(s):  
Supercompact Cardinal ◽  
Strong Limit ◽  
Limit Cardinal ◽  
Colour Class ◽  
Edge Colourings ◽  
Vertex Colouring ◽  
Edge Colouring

AbstractWe prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal μ, of cofinality ω, such that every μ+-chromatic graph X on μ+ has an edge colouring c of X into μ colours for which every vertex colouring g of X into at most μ many colours has a g-colour class on which c takes every value.The paper also contains some generalisations of the above statement in which μ+ is replaced by other cardinals > μ.

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