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.
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Bovykin, A. Resplendent models and \({\Sigma_1^1}\) -definability with an oracle. Arch. Math. Logic 47, 607–623 (2008). https://doi.org/10.1007/s00153-008-0100-8
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DOI: https://doi.org/10.1007/s00153-008-0100-8
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