A permutation model is a model of the axioms of set theory in which there is a non trivial automorphism of the set theoretic universe. Such models are used to show the consistency of the negation of the Axiom of Choice (AC).
A typical construction of a permutation model is done here. By we denote the axioms of minus the axiom of foundation. In particular we allow sets such that which we will call atoms. Let be an infinite set of atoms.
Define by induction on as follows:
Finally define . Then we have
For any we can assign a rank,
Let be the group of permutations of . For we extend to a permutation of by induction on by defining
and letting . Then permutes and fixes the well founded sets .
For all and any .
That is, is an -automorphism of . From this we can prove that and so
Also by induction on it is easy to show that
for all .
Let and define
is transitive and almost universal.
() If is transitive, almost universal and closed under Gödel Operations, then .
is closed under Gödel operations and so . The class is a permutation model. The set of atoms and furthermore:
Let be a one to one function. Then and so cannot be well ordered in .
Which proves the theorem:
which completes the proof that . In particular we have that .
|Date of creation||2013-03-22 14:46:48|
|Last modified on||2013-03-22 14:46:48|
|Last modified by||ratboy (4018)|