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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
##### Keywords:
DO models; Paris models
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##### References:
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