Hyperplanes in matroids and the axiom of choice

Marianne Morillon

Displaying similar documents to “Hyperplanes in matroids and the axiom of choice”

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...

Axioms weaker then $R_{0}$

J. Guia (1984)

Matematički Vesnik

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Set theories incorporating Hilbert's ε-symbol

T. B. Flannagan

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CONTENTS§ 1. Introduction ............................................................................................................ 5§ 2. The ε-calculus for є............................................................................................ 6§ 3. Reflection principles in e-set theories.............................................................. 6§ 4. [E]-elementary chains.......................................................................................... 11§ 5. Forcing...

Linear extenders and the Axiom of Choice

Marianne Morillon (2017)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the Axiom of Choice ZF, we prove that for every commutative field $𝕂$ , the following statement $𝐃_{𝕂}$ : “On every non null $𝕂$ -vector space, there exists a non null linear form” implies the existence of a “ $𝕂$ -linear extender” on every vector subspace of a $𝕂$ -vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case...

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"

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 the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product $2^{ℝ}$

Eleftherios Tachtsis (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets ( $A C^{W O}$ ) does not imply “the Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is countably compact” in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets...

Characterizations of some bitopological separation axioms in terms of $i j - θ$ - closure operator

S. K. Sen, J. N. Nandi, M. N. Mukherjee (1992)

Matematički Vesnik

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Axiom $T_{D}$ and the Simmons sublocale theorem

Jorge Picado, Aleš Pultr (2019)

Commentationes Mathematicae Universitatis Carolinae

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More precisely, we are analyzing some of H. Simmons, S. B. Niefield and K. I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_{D}$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation between sublocales and subspaces. S. B. Niefield and K. I. Rosenthal just mention this axiom in a remark about Simmons’ result....

$T_{i}$ - pairwise continuous functions and bitopological separation axioms

M. Jelić (1989)

Matematički Vesnik

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Covering Property Axiom $C P A_{c u b e}$ and its consequences

Krzysztof Ciesielski, Janusz Pawlikowski (2003)

Fundamenta Mathematicae

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We formulate a Covering Property Axiom $C P A_{c u b e}$ , which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨ f ₙ \in ℝ^{ℝ} ⟩_{n < ω}$ of Borel functions...

The $S_{α}$ separation axioms

J. Guia (1986)

Matematički Vesnik

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Embedding orders into the cardinals with $D C_{κ}$

Asaf Karagila (2014)

Fundamenta Mathematicae

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Jech proved that every partially ordered set can be embedded into the cardinals of some model of ZF. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $Z F + D C_{< κ}$ for any regular κ. We use this theorem to show that for all κ, the assumption of $D C_{κ}$ does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model....

Possibly there is no uniformly completely Ramsey null set of size $2^{ω}$

Andrzej Nowik (2002)

Colloquium Mathematicae

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We show that under the axiom $C P A_{c u b e}$ there is no uniformly completely Ramsey null set of size $2^{ω}$ . In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.

Essentially Incomparable Banach Spaces of Continuous Functions

Rogério Augusto dos Santos Fajardo (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We construct, under Axiom ♢, a family ${(C (K_{ξ}))}_{ξ < 2^{(2^{ω})}}$ of indecomposable Banach spaces with few operators such that every operator from $C (K_{ξ})$ into $C (K_{η})$ is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size $2^{ω}$ instead of $2^{(2^{ω})}$ .

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.......................................................................................................................................................

On pseudocompactness and related notions in ZF

Kyriakos Keremedis (2018)

Commentationes Mathematicae Universitatis Carolinae

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We study in ZF and in the class of $T_{1}$ spaces the web of implications/ non-implications between the notions of pseudocompactness, light compactness, countable compactness and some of their ZFC equivalents.