February 14
Ali Enayat,
University of Gothenburg
Models of set theory: extensions and dead-ends part II
This is a two-part talk concerning existence/non-existence of certain kinds of extensions of arbitrary models of ZF, with no regard to countability or well-foundedness of the models involved. The talk is based a recent preprint: arXiv:2406.14790v1. The results presented include the following two. In Theorem A below, N is said to be a conservative elementary extension of M if N is an elementary extension of M with the property that the intersection of every parametrically definable subset of N with M is parametrically definable in M.
Theorem A. Every model M of ZF with a definable global well-ordering has a conservative elementary extension N that contains an ordinal above all of the ordinals of M.
Theorem B. Every consistent extension of ZF has a model of power aleph_1 that has no end extension to a model of ZF.