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Permutation model
In mathematical set theory, a permutation model is a model of set theory with atoms (ZFA) constructed using a group of permutations of the atoms. A symmetric model is similar except that it is a model of ZF (without atoms) and is constructed using a group of permutations of a forcing poset. One application is to show the independence of the axiom of choice from the other axioms of ZFA or ZF.
Permutation models were introduced by and developed further by .
Symmetric models were introduced by Paul Cohen.
==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.

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