## Abstract.

A DO model (here also referred to a *Paris* model) is a model of set theory all of whose ordinals are first order definable in . 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 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 *κ* there is a Paris model of *ZF* of cardinality *κ* which has a nontrivial automorphism.

4. For a model *ZF*, is a prime model ⇒ is a Paris model and satisfies *AC*⇒ is a minimal model. Moreover, Neither implication reverses assuming Con(*ZF*).

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*Mathematics Subject Classification (2000):* 03C62, 03C50, Secondary 03H99

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Enayat, A. Models of set theory with definable ordinals.
*Arch. Math. Logic* **44, **363–385 (2005). https://doi.org/10.1007/s00153-004-0256-9

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DOI: https://doi.org/10.1007/s00153-004-0256-9

### Keywords

- Paris Model
- Mathematical Logic
- Minimal Model
- Prime Model
- Comprehensive Treatment