Models of set theory with definable ordinals

A DO model (here also referred to a Paris model) is a model M of set theory all of whose ordinals are first order definable in M. 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: 1. If T is a consistent completion of ZF+V =OD then T has continuummany countable nonisomorphic Paris models. 2. Every countable model of ZFC has a Paris generic extension. 3. If there is an uncountable well-founded model of ZFC, then for every infinite cardinal κ there is a Paris model of ZF of cardinality κ which has a nontrivial automorphism. 4. For a model M ZF ,M is a prime model ⇒M is a Paris model and satisfies AC ⇒M is a minimal model. Moreover, Neither implication reverses assuming Con(ZF).