Displaying similar documents to “Set-theoretic constructions of two-point sets”

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.

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.

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.

Internal and forcing models for the impredicative theory of classes

Rolando Chuaqui

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CONTENTSIntroduction............................................................................................................ 5I. Axiom system and elementary consequences........................................... 61. Axioms........................................................................................................................ 62. Definitions and elementary consequences........................................................ 9II. Principles of definitions by recursion..............................................................

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.