Permutation model: Difference between revisions

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In{{Short mathematical set theory, a '''permutation model''' is a modeldescription|Model of set theory with atoms (ZFA) constructed using a group of permutations of the atoms. }}

PermutationIn modelsmathematical were[[set introducedtheory]], bya {{harvs'''permutation model''' is a [[model (mathematical logic)|txtmodel]] of set theory with [[Atom (set theory)|last=Fraenkelatoms]] (ZFA) constructed using a [[permutation group|year=1922}}group]] of [[permutation]]s of the atoms. A '''symmetric model''' is similar except that it is a model of ZF (without atoms) and developedis furtherconstructed byusing {{harvs|txt|last=Mostowski|year=1938}}a group of permutations of a forcing [[poset]]. One application is to show the independence of the [[axiom of choice]] from variousthe other axioms inof setZFA theoryor with atomsZF.

Permutation models were introduced by {{harvs|txt|last=Fraenkel|year=1922}} and developed further by {{harvs|txt|last=Mostowski|year=1938}}.

Symmetric models were introduced by [[Paul Cohen (mathematician)|Paul Cohen]].

Paul COhen leter 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==

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*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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==Construction of filters on a group==

A filter on a group can be constructed from an invariant ideal on of the [[Boolean algebra]] of subsets of ''A'' containing all elements of ''A''. Here an ideal is a collection ''I'' of subsets of ''A'' closed under taking finite unions and subsets, and is called invariant if it is invariant under the action of the group ''G''. For each element ''S'' of the ideal one can take the subgroup of ''G'' consisting of all elements fixing every element ''S''. These subgroups generate a normal filter of ''G''.

==References==

*{{citation|last=Fraenkel|first= A.

|title=Der Begriff "definit" und die Unabhängigkeit des Auswahlaxioms| jfm =48.0199.02

|journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften|year= 1922|pages= 253–257 }}

*{{citation|first= Andrzej |last=Mostowski|title= Über den Begriff einer Endlichen Menge|year=1938|journal= Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III|volume=31|issue=8|pages=13–20}}

[[Category:Set theory]]