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Archive for Mathematical Logic

, Volume 47, Issue 6, pp 607–623 | Cite as

Resplendent models and \({\Sigma_1^1}\) -definability with an oracle

  • Andrey BovykinEmail author
Article

Abstract

In this article we find some sufficient and some necessary \({\Sigma^1_1}\) -conditions with oracles for a model to be resplendent or chronically resplendent. The main tool of our proofs is internal arguments, that is analogues of classical theorems and model-theoretic constructions conducted inside a model of first-order Peano Arithmetic: arithmetised back-and-forth constructions and versions of the arithmetised completeness theorem, namely constructions of recursively saturated and resplendent models from the point of view of a model of arithmetic. These internal arguments are used in conjunction with Pabion’s theorem that ensures that certain oracles are coded in a sufficiently saturated model of arithmetic. Examples of applications are provided for the theories of dense linear orders and of discrete linear orders. These results are then generalised to other ω-categorical theories and theories with a unique countable recursively saturated model.

Keywords

Resplendent model Arithmetised completeness theorem Pabion’s theorem Chronically resplendent model ω-categorical theory Linear order Dense linear order Discrete linear order Model theory Expansion of a structure Recursively saturated model 

Mathematics Subject Classification (2000)

Primary 03B10 03C50 03C57 03C62 Secondary 03C07 03C30 03C35 03C52 03C64 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsBristol UniversityBristolUK

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