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Models of set theory with definable ordinals

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+VOD, 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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Correspondence to Ali Enayat.

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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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Keywords

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