Archive for Mathematical Logic 44 (3):363-385 (2004)

Authors
Ali Enayat
University of Gothenburg
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)
Keywords Mathematics   Mathematics, general   Algebra   Mathematical Logic and Foundations
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Reprint years 2005
DOI 10.1007/s00153-004-0256-9
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References found in this work BETA

Consistency Results About Ordinal Definability.Kenneth McAloon - 1971 - Annals of Pure and Applied Logic 2 (4):449.
Leibnizian Models of Set Theory.Ali Enayat - 2004 - Journal of Symbolic Logic 69 (3):775-789.
New Set-Theoretic Axioms Derived From a Lean Metamathematics.Jan Mycielski - 1995 - Journal of Symbolic Logic 60 (1):191-198.

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Citations of this work BETA

Pointwise Definable Models of Set Theory.Joel David Hamkins, David Linetsky & Jonas Reitz - 2013 - Journal of Symbolic Logic 78 (1):139-156.
Model Theory of the Regularity and Reflection Schemes.Ali Enayat & Shahram Mohsenipour - 2008 - Archive for Mathematical Logic 47 (5):447-464.
Minimum Models of Second-Order Set Theories.Kameryn J. Williams - 2019 - Journal of Symbolic Logic 84 (2):589-620.

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