Permutation model: Difference between revisions

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In mathematical [[set theory]], a '''permutation model''' is a [[model (mathematical logic)|model]] of set theory with atoms (ZFA) constructed using a group of [[permutation]]s 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 {{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.
Symmetric models were introduced by [[Paul Cohen (mathematician)|Paul Cohen]].
 
[[Paul Cohen (mathematician)|Paul Cohen]] later used similar ideas to construct models of ZF not satisfying the axiom of choice, by replacing a group acting on a set of atoms by a group acting on a forcing poset.
 
==Construction of permutation models==