Models of set theory with definable ordinals

Abstract

A DO model (here also referred to a Paris model) is a model Open image in new window of set theory all of whose ordinals are first order definable in Open image in new window. Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our results include the following: