Linear forms and axioms of choice

Marianne Morillon

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 $2^{ℝ}$ , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product ${[0, 1]}^{ℝ}$ 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 $ℵ_{ω}$ is strong limit and $2^{ℵ_{ω}} > ℵ_{ω_{1}}$ 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 $(X, ℱ)$ , $X$ a compact Hausdorff space and $ℱ$ a filter of dense open subsets of $X$ . A morphism $f (Y, 𝒢) \to (X, ℱ)$ is a continuous function $f Y \to X$ for which $f^{- 1} (F) \in 𝒢$ whenever $F \in ℱ$ . 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 $(V_{α}, \in)$ 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 $𝐟 : I \to ℝ^{k},$ where $I$ is a real interval and $k \in ℕ$ , a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean $M : I^{2} \to I$ such that $𝐟 (x) - 𝐟 (y) = (x - y) 𝐟^{'} (M (x, y)), x, y \in I,$ are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.