Displaying similar documents to “On the Leibniz-Mycielski axiom in set theory”

Hyperplanes in matroids and the axiom of choice

Marianne Morillon (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show that in set theory without the axiom of choice ZF, the statement sH: “Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it” implies AC fin , the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC fin in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?

The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brelot

Peter Loeb, Bertram Walsh (1965)

Annales de l'institut Fourier

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Dans l’axiomatique des fonctions harmoniques de Brelot, où l’axiome 3 (de convergence) peut être appelé principe de Harnack, on démontre ici pour les fonctions harmoniques > 0 dans un domaine ω valant 1 en x 0 ∈ ω , la propriété d’égale continuité en x 0 qui peut se traduire par des “inégalités de Harnack”. Cela avait été établi par Mokobodzki grâce à l’hypothèse d’une base dénombrable d’ouverts, qui est évitée ici en utilisant le théorème d’Éberlein-Smulian.

Complexity of the axioms of the alternative set theory

Antonín Sochor (1993)

Commentationes Mathematicae Universitatis Carolinae

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If is a complete theory stronger than Fin such that axiom of extensionality for classes + + ( ∃ X ) Φ i is consistent for 1 ≤ i ≤ k (each alone), where Φ i are normal formulae then we show + ( ∃ X ) Φ 1 + ⋯ + ( ∃ X ) Φ k + scheme of choice is consistent. As a consequence we get: there is no proper Δ 1 -formula in + scheme of choice. Moreover the complexity of the axioms of is studied, e.gẇe show axiom of extensionality is Π 1 -formula, but not Σ 1 -formula and furthermore prolongation axiom, axioms of choice and cardinalities...

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 → 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 a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

Horst Herrlich, Paul Howard, Eleftherios Tachtsis (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice ( AC ), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y → X.

Inaccessible cardinals without the axiom of choice

Andreas Blass, Ioanna M. Dimitriou, Benedikt Löwe (2007)

Fundamenta Mathematicae

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We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.

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.

Stranger things about forcing without AC

Martin Goldstern, Lukas D. Klausner (2020)

Commentationes Mathematicae Universitatis Carolinae

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Typically, set theorists reason about forcing constructions in the context of Zermelo--Fraenkel set theory (ZFC). We show that without the axiom of choice (AC), several simple properties of forcing posets fail to hold, one of which answers Miller's question from the work: Arnold W. Miller, {Long Borel hierarchies}, MLQ Math. Log. Q. {54} (2008), no. 3, 307--322.

[unknown]

M. Jelić (1990)

Matematički Vesnik

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