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Displaying similar documents to “Cardinality of the sets of all bijections, injections and surjections”

Lusin sequences under CH and under Martin's Axiom

Uri Abraham, Saharon Shelah (2001)

Fundamenta Mathematicae

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Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.

Axioms which imply GCH

Jan Mycielski (2003)

Fundamenta Mathematicae

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We propose some new set-theoretic axioms which imply the generalized continuum hypothesis, and we discuss some of their consequences.

Set-theoretic constructions of two-point sets

Ben Chad, Robin Knight, Rolf Suabedissen (2009)

Fundamenta Mathematicae

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A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the...

New axioms in set theory

Giorgio Venturi, Matteo Viale (2018)

Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana

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In this article we review the present situation in the foundations of set theory, discussing two programs meant to overcome the undecidability results, such as the independence of the continuum hypothesis; these programs are centered, respectively, on forcing axioms and Woodin's V = Ultimate-L conjecture. While doing so, we briefly introduce the key notions of set theory.

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 partial orderings having precalibre-ℵ₁ and fragments of Martin's axiom

Joan Bagaria, Saharon Shelah (2016)

Fundamenta Mathematicae

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We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-ℵ₁, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for σ-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for σ-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer...

Cylinder problem

K. Ciesielski, F. Galvin (1987)

Fundamenta Mathematicae

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