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.

Regressive partitions and Borel diagonalization

1989 ◽  
Vol 54 (2) ◽  
pp. 540-552 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Set Theory ◽  
Natural Generalization ◽  
Large Cardinal ◽  
Unit Interval ◽  
Combinatorial Proof ◽  
Consistency Strength ◽  
Several Variables ◽  
Borel Measurable ◽  
Mahlo Cardinals ◽  
Level Analysis

Several rather concrete propositions about Borel measurable functions of several variables on the Hilbert cube (countable sequences of reals in the unit interval) were formulated by Harvey Friedman [F1] and correlated with strong set-theoretic hypotheses. Most notably, he established that a “Borel diagonalization” proposition P is equivalent to: for any a ⊆ co and n ⊆ ω there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. In later work (see the expository Stanley [St] and Friedman [F2]), Friedman was to carry his investigations further into propositions about spaces of groups and the like, and finite propositions. He discovered and analyzed mathematical propositions which turned out to have remarkably strong consistency strength in terms of large cardinal hypotheses in set theory.In this paper, we refine and extend Friedman's work on the Borel diagonalization proposition P. First, we provide more combinatorics about regressive partitions and n-Mahlo cardinals and extend the approach to the context of the Erdös cardinals In passing, a combinatorial proof of a well-known result of Silver about these cardinals is given. Incorporating this work and sharpening Friedman's proof, we then show that there is a level-by-level analysis of P which provides for each n ⊆ ω a proposition almost equivalent to: for any a ⊆ co there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. Finally, we use the combinatorics to bracket a natural generalization Sω of P between two large cardinal hypotheses.

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.

ON ASTUMIAN'S PARADOX

2005 ◽  
Vol 05 (01) ◽  
pp. L109-L125
Author(s):  
EHRHARD BEHRENDS
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Transition Probabilities ◽  
Natural Generalization ◽  
The Other ◽  
Finite Markov Chain ◽  
Fair Coin ◽  
Winning Probability ◽  
Starting Position ◽  
Absorbing States

An Astumian game is defined by a finite Markov chain with state space S with precisely two absorbing states, the winning and the losing state. The other states are transient, one of them is the starting position. The game is said to be losing (respectively fair respectively winning) if the probability to be absorbed at the winning state is smaller than 0.5 (respectively = 0.5 respectively larger than 0.5). Astumian's paradox states that there are losing games on the same state space S a stochastic mixture of which is winning. (By "stochastic mixture" we mean that in each step one decides with the help of a fair coin whether to use the transition probabilities of the first or the second game.) Most of our results concern fair games. Mixtures are systematically investigated. Rather surprisingly, the winning probability of the mixture of fair games can be arbitrarily close to zero (or to one). Even more counter-intuitive are examples of definitely losing games (this means that the winning probability is exactly zero) such that the winning probability of the mixture is arbitrarily close to one. We show, however, that such extreme examples are possible only if one tolerates huge running times of the game. As a natural generalization one can also consider arbitrary mixtures: the fair coin is replaced by a biased one, with probability λ respectively 1-λ one plays with the first respectively the second game. It turns out that fair games exist such that — depending on the choice of λ — the λ-mixture can be fair, losing or winning.

scholarly journals LOCAL CLUB CONDENSATION AND L-LIKENESS

2015 ◽  
Vol 80 (4) ◽  
pp. 1361-1378
Author(s):  
PETER HOLY ◽  
PHILIP WELCH ◽  
LIUZHEN WU
Keyword(s):  
Regular Cardinal ◽  
Large Cardinal ◽  
Ground Model ◽  
The Third ◽  
Chang’S Conjecture ◽  
Chang's Conjecture ◽  
Generalized Condensation

AbstractWe present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model. We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ+ is consistent with ¬☐κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω2 is regular, CC(κ+) - Chang’s Conjecture at κ+ - is consistent with Local Club Condensation at κ+, both under suitable large cardinal consistency assumptions.

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 Non decay of the total energy for the wave equation with the dissipative term of spatial anisotropy

2004 ◽  
Vol 174 ◽  
pp. 115-126 ◽  
Author(s):  
Mishio Kawashita ◽  
Hideo Nakazawa ◽  
Hideo Soga
Keyword(s):  
Wave Equation ◽  
Total Energy ◽  
Natural Generalization ◽  
The Other ◽  
Asymptotic Solutions ◽  
Half Line ◽  
Decay Estimates ◽  
Uniform Decay ◽  
Spatial Anisotropy ◽  
Dissipative Term

AbstractWe consider the behavior of the total energy for the wave equation with the dissipative term. When the dissipative term works well uniformly in every direction, several authors obtain uniform decay estimates of the total energy. On the other hand, if the dissipative term is small enough uniformly in every direction, it is known that there exists a solution whose total energy does not decay. We examine the case that the dissipative term vanishes only in a neighborhood of a half-line. We introduce a uniform decay property, which is a natural generalization of the uniform decay estimates, and show that this property does not hold in our case. We prove this by constructing asymptotic solutions supported in the place where the dissipative term vanishes.

scholarly journals Combined Maximality Principles up to large cardinals

2009 ◽  
Vol 74 (3) ◽  
pp. 1015-1046 ◽  
Author(s):  
Gunter Fuchs
Keyword(s):  
Regular Cardinal ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Consistency Strength ◽  
Regular Cardinals ◽  
Maximality Principles

AbstractThe motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.

Combinatorics on large cardinals

1992 ◽  
Vol 57 (2) ◽  
pp. 617-643 ◽  
Author(s):  
Carlos H. Montenegro E.
Keyword(s):  
Regular Cardinal ◽  
Ordered Set ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Combinatorial Properties ◽  
A Chain ◽  
Linearly Ordered Set ◽  
Normal Ideals ◽  
The Ideal

Our framework is ZFC, and we view cardinals as initial ordinals. Baumgartner ([Bal] and [Ba2]) studied properties of large cardinals by considering these properties as properties of normal ideals and not as properties of cardinals alone. In this paper we study these combinatorial properties by defining operations which take as input one or more ideals and give as output an ideal associated with a large cardinal property. We consider four operations T, P, S and C on ideals of a regular cardinal κ, and study the structure of the collection of subsets they give, and the relationships between them.The operation T is defined using combinatorial properties based on trees 〈X, <T〉 on subsets X ⊆ κ (where α <T β → α < β). Given an ideal I, consider the property *: “every tree on κ with every branching set in I has a branch of size κ” (where a branching set is a maximal set with the same set of <T-predecessors, and a chain is a maximal <T-linearly ordered set; for definitions see §2). Now consider the collection T(I) of all subsets of κ that do not satisfy * (see Definition 2.2 and the introduction to §5). The operation T provides us with the large cardinal property (whether κ ∈ T(I) or not) and it also provides us with the ideal associated with this large cardinal property (namely T(I)); in general, we obtain different notions depending on the ideal I.

On the existence of strong chains in ℘(ω1)/Fin

1998 ◽  
Vol 63 (3) ◽  
pp. 1055-1062 ◽  
Author(s):  
Piotr Koszmider
Keyword(s):  
The Other ◽  
Disjoint Family ◽  
Other Hand ◽  
Chang’S Conjecture ◽  
Almost Disjoint ◽  
Chang's Conjecture ◽  
Almost Disjoint Family

Abstract(Xα: α < ω2) ⊂ ℘(ω1) is a strong chain in ℘(ω1)/Fin if and only if Xβ – Xα is finite and Xα – Xβ is uncountable for each β < α < ω1. We show that it is consistent that a strong chain in ℘(ω1) exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in ℘(ω1) but no strong chain exists: is used to construct a c.c.c forcing that adds a strong chain and Chang's Conjecture to prove that there is no strong chain.

THE DYNAMICAL THEORY OF MAGNETIC MONOPOLES

1952 ◽  
Vol 30 (3) ◽  
pp. 218-225 ◽  
Author(s):  
S. Shanmugadhasan
Keyword(s):  
Electromagnetic Field ◽  
Equations Of Motion ◽  
Natural Generalization ◽  
Magnetic Monopoles ◽  
Dynamical Theory ◽  
The Other ◽  
Field Tensor ◽  
Physical Content ◽  
Electromagnetic Field Tensor ◽  
Set Up

The theory of electric charges and magnetic monopoles has been set up by Dirac by expressing the electromagnetic field tensor in terms of one four-potential and of the variables describing the strings attached to each magnetic mono-pole. In this reformulation of Dirac's theory the field tensor is expressed in terms of two four-potentials, one corresponding to charges and the other to monopoles, and the action principle for the equations of motion is set up in terms of the two four-potentials and of the tensors dual to them. Thus there is formal symmetry as far as is possible in the treatment of the charges and the monopoles. Also the mathematics is direct and neat. Though the physical content is the same as that of Dirac, a natural generalization of the Fermi form of electrodynamics subject to the restriction that the same particle cannot have both charge and monopole is obtained here.

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