Displaying similar documents to “The 'World's Simplest Axiom of Choice' Fails.”

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

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.

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...

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M. Jelić (1990)

Matematički Vesnik

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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.