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Permutation models were introduced by {{harvs|txt|last=Fraenkel|year=1922}} and developed further by {{harvs|txt|last=Mostowski|year=1938}}. One application is to show the independence of the [[axiom of choice]] from various other axioms in set theory with atoms.
==Construction of permutation models==
Suppose that ''A'' is a set of atoms, and ''G'' is a group of permutations of ''A''. A '''normal filter''' of ''G'' is a collection ''F'' of subgroups of ''G'' such that
*''G'' is in ''F''
*The intersection of two elements of ''F'' is in ''F''
*Any subgroup containing an element of ''F'' is in ''F''
*Any conjugate of an element of ''F'' is in ''F''
*The subgroup fixing any element of ''A'' is in ''F''.
If ''V'' is a model of ZFA with ''A'' the set of atoms, then an element of ''V'' is called symmetric if the subgroup fixing it is in ''F'', and is called hereditarily symmetric if it and all elements of its transitive closure are symmetric. The '''permutation model''' consists of all hereditarily symmetric elements, and is a model of ZFA.
==References==
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