On the set-theoretic strength of the n-compactness of generalized Cantor cubes

Paul Howard; Eleftherios Tachtsis

Displaying similar documents to “On the set-theoretic strength of the n-compactness of generalized Cantor cubes”

Remarks on the Stone Spaces of the Integers and the Reals without AC

Horst Herrlich, Kyriakos Keremedis, Eleftherios Tachtsis (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product $2^{(X)}$ , where 2 is 2 = 0,1 with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X = ω,ℝ. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

Tychonoff Products of Two-Element Sets and Some Weakenings of the Boolean Prime Ideal Theorem

Kyriakos Keremedis (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be an infinite set, and (X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of (X) can be extended to an ultrafilter. UF(X): (X) has a free ultrafilter. We will show in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI(ω). (ii) The Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product ${[0, 1]}^{ℝ}$ is compact. (iv) In a...

On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall, Kyriakos Keremedis (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product $2^{(ω)}$ the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of $2^{(ω)}$ to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean...

The reaping and splitting numbers of nice ideals

Rafał Filipów (2014)

Colloquium Mathematicae

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We examine the splitting number (B) and the reaping number (B) of quotient Boolean algebras B = (ω)/ℐ where ℐ is an $F_{σ}$ ideal or an analytic P-ideal. For instance we prove that under Martin’s Axiom ((ω)/ℐ) = for all $F_{σ}$ ideals ℐ and for all analytic P-ideals ℐ with the BW property (and one cannot drop the BW assumption). On the other hand under Martin’s Axiom ((ω)/ℐ) = for all $F_{σ}$ ideals and all analytic P-ideals ℐ (in this case we do not need the BW property). We also provide applications...

A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem

Miroslav Repický (2015)

Commentationes Mathematicae Universitatis Carolinae

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We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.

The gap between I₃ and the wholeness axiom

Paul Corazza (2003)

Fundamenta Mathematicae

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∃κI₃(κ) is the assertion that there is an elementary embedding $i : V_{λ} \to V_{λ}$ with critical point below λ, and with λ a limit. The Wholeness Axiom, or WA, asserts that there is a nontrivial elementary embedding j: V → V; WA is formulated in the language ∈,j and has as axioms an Elementarity schema, which asserts that j is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by j; and includes every instance of the Separation schema for j-formulas. Because no instance...

On the Compactness and Countable Compactness of $2^{ℝ}$ in ZF

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " $2^{ℝ}$ is countably compact" and " $2^{ℝ}$ is compact"

An axiomatics of non-Desarguean geometry based on the half-plane as the primitive notion

A. Śniatycki

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CONTENTSIntroduction................................................................................................................................................. 5PART I1. Axioms of Boolean algebra................................................................................................................. 62. Half-planes and their axioms.............................................................................................................. 73. The line.......................................................................................................................................................

Kelley's specialization of Tychonoff's Theorem is equivalent to the Boolean Prime Ideal Theorem

Eric Schechter (2006)

Fundamenta Mathematicae

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The principle that "any product of cofinite topologies is compact" is equivalent (without appealing to the Axiom of Choice) to the Boolean Prime Ideal Theorem.

Axioms weaker then $R_{0}$

J. Guia (1984)

Matematički Vesnik

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$\forall_{n}$ -theories of Boolean algebras

Jan Waszkiewicz (1974)

Colloquium Mathematicae

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Making sense of capitulation: reciprocal primes

David Folk (2016)

Acta Arithmetica

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Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and $a_{}$ any generator of the principal ideal $^{ℓ}$ . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $G a l (K (\sqrt[ℓ]{a_{}}) / K)$ for every choice of $a_{}$ . We then show that becomes principal in L if and only if every reciprocal prime is not...

Boolean algebras, splitting theorems, and $Δ_{2}^{0}$ sets

Michael Morley, Robert Soare (1975)

Fundamenta Mathematicae

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The Tychonoff product theorem for compact Hausdorff spaces does not imply the axiom of choice: A new Proof. equivalent propositions.

B. R. Salinas, F. Bombal (1973)

Collectanea Mathematica

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Generalised irredundance in graphs: Nordhaus-Gaddum bounds

Ernest J. Cockayne, Stephen Finbow (2004)

Discussiones Mathematicae Graph Theory

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For each vertex s of the vertex subset S of a simple graph G, we define Boolean variables p = p(s,S), q = q(s,S) and r = r(s,S) which measure existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p,q,r) may be considered as a compound existence property of S-pns. The subset S is called an f-set of G if f = 1 for all s ∈ S and the class of f-sets of G is denoted by $Ω_{f} (G)$ . Only 64 Boolean functions f can produce different classes $Ω_{f} (G)$ , special cases...

Partial choice functions for families of finite sets

Eric J. Hall, Saharon Shelah (2013)

Fundamenta Mathematicae

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Let m ≥ 2 be an integer. We show that ZF + “Every countable set of m-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure...

Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

Carlos Matheus, Carlos G. Moreira, Enrique R. Pujals (2013)

Annales scientifiques de l'École Normale Supérieure

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We consider a family of planar systems introduced in 1991 by Benedicks and Carleson as a toy model for the dynamics of the so-called Hénon maps. We show that Smale’s Axiom A property is $C^{1}$ -dense among the systems in this family, despite the existence of $C^{2}$ -open subsets (closely related to the so-called Newhouse phenomena) where Smale’s Axiom A is violated. In particular, this provides some evidence towards Smale’s conjecture that Axiom A is a $C^{1}$ -dense property among surface diffeomorphisms. The...

On $K$ -Boolean Rings

W. B. Vasantha Kandasamy (1992)

Publications du Département de mathématiques (Lyon)

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Wild primes of a self-equivalence of a number field

Alfred Czogała, Beata Rothkegel (2014)

Acta Arithmetica

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Let K be a number field. Assume that the 2-rank of the ideal class group of K is equal to the 2-rank of the narrow ideal class group of K. Moreover, assume K has a unique dyadic prime and the class of is a square in the ideal class group of K. We prove that if ₁,...,ₙ are finite primes of K such that ∙ the class of $_{i}$ is a square in the ideal class group of K for every i ∈ 1,...,n, ∙ -1 is a local square at $_{i}$ for every nondyadic $_{i} \in ₁, . . ., ₙ$ , then ₁,...,ₙ is the wild set of some self-equivalence...

Laslett’s transform for the Boolean model in $ℝ^{d}$

Rostislav Černý (2006)

Kybernetika

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Consider a stationary Boolean model $X$ with convex grains in $ℝ^{d}$ and let any exposed lower tangent point of $X$ be shifted towards the hyperplane $N_{0} = {x \in ℝ^{d} : x_{1} = 0}$ by the length of the part of the segment between the point and its projection onto the $N_{0}$ covered by $X$ . The resulting point process in the halfspace (the Laslett’s transform of $X$ ) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie...

Iterated Boolean random varieties and application to fracture statistics models

Dominique Jeulin (2016)

Applications of Mathematics

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Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in $ℝ^{2}$ and $ℝ^{3}$ and on random planes in $ℝ^{3}$ . The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set $K$ and the Choquet capacity $T (K)$ are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical...