In mathematical
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, a permutation model is a
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
of set theory with
atoms
Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
(ZFA) constructed using a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
of
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
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
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
. One application is to show the independence of the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
from the other axioms of ZFA or ZF.
Permutation models were introduced by and developed further by .
Symmetric models were introduced by
Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a F ...
.
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.
Construction of filters on a group
A filter on a group can be constructed from an invariant ideal on of the
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
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, 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
Set theory