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.

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]).

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.

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.

Interpolation theorems for Lk,k2+

1978 ◽  
Vol 43 (3) ◽  
pp. 535-549 ◽  
Author(s):  
Ruggero Ferro
Keyword(s):  
Natural Deduction ◽  
Interpolation Theorem ◽  
Positive Answer ◽  
Strong Limit ◽  
First Order ◽  
Limit Cardinal ◽  
Saturated Models ◽  
Order Language ◽  
Craig’S Interpolation Theorem

Chang, in [1], proves an interpolation theorem (Theorem I, remark b)) for a first-order language. The proof of Chang's theorem uses essentially nonsimple devices, like special and ω1-saturated models.In remark e) in [1], Chang asks if there is a simpler proof of his Theorem I.In [1], Chang proves also another interpolation theorem (Theorem II), which is not an extension of his Theorem I, but extends Craig's interpolation theorem to Lα+,ω languages with interpolant in Lα+,α where α is a strong limit cardinal of cofinality ω.In remark k) in [1], Chang asks if there is a generalization of both Theorems I and II in [1], or at least a generalization of both Theorem I in [1] and Lopez-Escobar's interpolation theorem in [7].Maehara and Takeuti, in [8], show that there is a completely different proof of Chang's interpolation Theorem I as a consequence of their interpolation theorems. The proofs of these theorems of Maehara and Takeuti are proof theoretical in character, involving the notion of cut-free natural deduction, and it uses devices as simple as those needed for the usual Craig's interpolation theorem. Hence this can be considered as a positive answer to Chang's question in remark e) in [1].

Undefinability of κ-well-orderings in L∞κ

1997 ◽  
Vol 62 (3) ◽  
pp. 999-1020 ◽  
Author(s):  
Juha Oikkonen
Keyword(s):  
Strong Limit ◽  
Limit Cardinal

AbstractWe prove that the class of trees with no branches of cardinality ≤ κ is not RPC definable in L∞κ when κ is regular. Earlier such a result was known for under the assumption κ<κ = κ. Our main result is actually proved in a stronger form which covers also L∞κ (and makes sense there) for every strong limit cardinal λ < κ of cofinality κ.

Chang's conjecture and powers of singular cardinals

1977 ◽  
Vol 42 (2) ◽  
pp. 272-276 ◽  
Author(s):  
Menachem Magidor
Keyword(s):  
Strong Limit ◽  
Elementary Substructure ◽  
Limit Cardinal ◽  
Countable Type ◽  
Chang’S Conjecture ◽  
Singular Cardinals ◽  
Nontrivial Ideal ◽  
Chang's Conjecture ◽  
Image Position ◽  
The Given

In [2] Galvin and Hajnal showed, as a corollary to a more general result, that if , is a strong limit cardinal, then . They established similar bounds for powers of singular cardinals of cofinality greater than ω. Jech and Prikry in [3] showed that the Galvin-Hajnal bound can be improved if we assume that ω1 carries an ω2 saturated ω1 complete, nontrivial ideal. (See [7] for definitions), namely: under the given assumption provided is a strong limit cardinal.In this paper we show that the same conclusion can be derived from Chang's Conjecture (see below) which is, at least consistencywise, a weaker assumption than the existence of an ω2 saturated ideal on ω1. We do not know if assumptions like these are necessary for obtaining the result.Our notations and terminology should be understood by any reader acquainted with set theory. Chang's Conjecture is the following model theoretic assumption introduced by C. C. Chang:which is deciphered as follows: Every structure 〈A, R,…〉 in a countable type where ∣A∣ = ω2, R ⊆ A, ∣R∣ = ω1 has an elementary substructure: 〈A′,R′,…〉 where ∣A′∣ = ω1 and ∣R′∣ = ω0. The consistency of Chang's Conjecture modulo the existence of Ramsey cardinals is claimed in [5].

Filters, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller Theorem

2000 ◽  
Vol 65 (1) ◽  
pp. 259-271 ◽  
Author(s):  
Saharon Shelah ◽  
Lee J. Stanley
Keyword(s):  
Subsequent Paper ◽  
Strong Limit ◽  
Limit Cardinal ◽  
Singular Cardinals ◽  
Different Types

AbstractWe present two different types of models where, for certain singular cardinals λ of uncountable cofinality, λ → (λ, ω + 1)2, although λ is not a strong limit cardinal, We announce, here, and will present in a subsequent paper, [7], that, for example, consistently, and consistently, .

Nontame mouse from the failure of square at a singular strong limit cardinal

2014 ◽  
Vol 14 (01) ◽  
pp. 1450003 ◽  
Author(s):  
Grigor Sargsyan
Keyword(s):  
Core Model ◽  
Strong Limit ◽  
Inductive Step ◽  
Limit Cardinal ◽  
The Core

Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216(1) (2007) 89–117] and Steel [PFA implies AD L(ℝ), J. Symbolic Logic70(4) (2005) 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L(ℝ) of the core model induction that is aimed at getting a model of ADℝ + "Θ is regular" from the failure of square at a singular strong limit cardinal or PFA.

scholarly journals Sigma-Prikry forcing I: The Axioms

2020 ◽  
pp. 1-34
Author(s):  
Alejandro Poveda ◽  
Assaf Rinot ◽  
Dima Sinapova
Keyword(s):  
Iteration Scheme ◽  
Finite Collection ◽  
Strong Limit ◽  
Prikry Forcing ◽  
Supercompact Cardinals ◽  
Limit Cardinal ◽  
Singular Cardinals ◽  
Stationary Set ◽  
Stationary Subsets

Abstract We introduce a class of notions of forcing which we call $\Sigma $ -Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma $ -Prikry. We show that given a $\Sigma $ -Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set T, there exists a corresponding $\Sigma $ -Prikry poset that projects to $\mathbb P$ and kills the stationarity of T. Then, in a sequel to this paper, we develop an iteration scheme for $\Sigma $ -Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $\kappa $ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which $\kappa $ remains a strong limit cardinal, every finite collection of stationary subsets of $\kappa ^+$ reflects simultaneously, and $2^\kappa =\kappa ^{++}$ .

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