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Models of set theory with definable ordinals. (English) Zbl 1068.03032
Summary: A DO model (here also referred to as a Paris model) is a model \(\mathfrak M\) of set theory all of whose ordinals are first-order definable in \(\mathfrak M\). 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 \(\mathbf V = \mathbf{OD}\). Here we provide a comprehensive treatment of Paris models. Our results include the following:
1. If \(T\) is a consistent completion of \(\text{ZF}+\mathbf V \not= \mathbf{OD}\), then \(T\) has continuum-many 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 \(\kappa\) there is a Paris model of ZF of cardinality \(\kappa\) which has a nontrivial automorphism.
4. For a model \(\mathfrak M \vDash \text{ZF}\), \(\mathfrak M\) is a prime model \(\Rightarrow \mathfrak M\) is a Paris model and satisfies \(\text{AC} \Rightarrow \mathfrak M\) is a minimal model. Moreover, neither implication reverses assuming Con(ZF).

MSC:
03C62 Models of arithmetic and set theory
03E35 Consistency and independence results
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