Generic extensions of permutation models of set theory.

Abstract

This thesis gives some general results about generalized Fraenkel-Mostowski-Specker (FMS) models and symmetric models, which are models of Zermelo-Fraenkel (ZF) set theory in which the Axiom of Choice (AC) does not hold. A new characterization of FMS models is given in terms of their generic extensions. If <italic>M</italic> is a model of ZFA (ZF set theory with atoms) +AC and <italic>N</italic> is an inner model of <italic>M</italic>, and if <italic> M</italic> and <italic>N</italic> have the same pure sets (sets with no atoms in their transitive closure) then <italic>N</italic> is an FMS submodel of <italic> M</italic> if and only if <italic>M</italic> is a generic extension of <italic> N</italic> by some almost homogeneous notion of forcing. We generalize the usual construction of FMS submodels of <italic>M</italic> by using a group of permutations of the atoms that is not necessarily in <italic>M</italic> and call the general construction a permutation submodel. <italic> N</italic> can be obtained as a permutation submodel by a group that is definable using only parameters from <italic>M</italic> in a generic extension of <italic> M</italic> by some almost homogeneous notion of forcing if and only if <italic> M</italic> is a generic extension of <italic>N</italic>. A new characterization of symmetric models in terms of permutation models is given. The kernel of a model of ZFA is the class of pure sets in the model. Symmetric models are precisely those models of ZF which are kernels of generic extensions of permutation models. We also investigate necessary and sufficient conditions under which certain standard notions of forcing over a (generalized) FMS model add well-orderings of or well-ordered subsets of sets in the ground model.