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An extension of a permutative model of set theory. (English) Zbl 1274.03073

Fraenkel-Mostowski permutation models of set theory were introduced to show the independence of the axiom of choice from Zermelo-Fraenkel set theory with atoms. The authors discuss a generalization of Fraenkel-Mostowski (FM) models, called extended Fraenkel-Mostowski (EFM) models, in which the finite support property of FM models (saying that, given any set \(x\), there is a finite set \(S\) of atoms such that any permutation of the atoms which fixes \(S\) pointwise also fixes \(x\)) is replaced by the weaker requirement that any subset of the atoms is either finite or cofinite. They show that EFM models still share some properties with FM models, e.g., the permutation group of the atoms is a torsion group and all of its finitely generated subgroups are finite.

MSC:

03E30 Axiomatics of classical set theory and its fragments
03E25 Axiom of choice and related propositions
20B07 General theory for infinite permutation groups
20E25 Local properties of groups
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References:

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