Models of set theory with definable ordinals

Archive for Mathematical Logic 44 (3):363-385 (2005)
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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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Citations of this work

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References found in this work

Leibnizian models of set theory.Ali Enayat - 2004 - Journal of Symbolic Logic 69 (3):775-789.
Consistency results about ordinal definability.Kenneth McAloon - 1971 - Annals of Mathematical Logic 2 (4):449.
New set-theoretic axioms derived from a lean metamathematics.Jan Mycielski - 1995 - Journal of Symbolic Logic 60 (1):191-198.

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