Displaying similar documents to “α-Properness and Axiom A”

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.

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...

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.

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

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M. Jelić (1990)

Matematički Vesnik

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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.