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Displaying similar documents to “Linear forms and axioms of choice”

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 , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product [ 0 , 1 ] is compact. (iv) In a...

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.

Another proof of a result of Jech and Shelah

Péter Komjáth (2013)

Czechoslovak Mathematical Journal

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Shelah’s pcf theory describes a certain structure which must exist if ω is strong limit and 2 ω > ω 1 holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that...

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

Paul Howard, Eleftherios Tachtsis (2016)

Fundamenta Mathematicae

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We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product 2 X , where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel...

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...

Monomorphisms in spaces with Lindelöf filters

Richard N. Ball, Anthony W. Hager (2007)

Czechoslovak Mathematical Journal

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𝐒𝐩𝐅𝐢 is the category of spaces with filters: an object is a pair ( X , ) , X a compact Hausdorff space and a filter of dense open subsets of X . A morphism f ( Y , 𝒢 ) ( X , ) is a continuous function f Y X for which f - 1 ( F ) 𝒢 whenever F . This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically...

On the Leibniz-Mycielski axiom in set theory

Ali Enayat (2004)

Fundamenta Mathematicae

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Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that ( V α , ) satisfies φ(x) ∧¬ φ(y). We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows: 1. In the presence of ZF, the following are equivalent: (a)...

Mean-value theorem for vector-valued functions

Janusz Matkowski (2012)

Mathematica Bohemica

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For a differentiable function 𝐟 : I k , where I is a real interval and k , a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean M : I 2 I such that 𝐟 ( x ) - 𝐟 ( y ) = ( x - y ) 𝐟 ' ( M ( x , y ) ) , x , y I , are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.