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Resplendent models and Σ 1 1-definability with an oracle

By Andrey Bovykin


In this article we find some sufficient and some necessary Σ 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 backand-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 DLO (of dense linear orders) and DIS (of discrete linear orders). These results are then generalised to other ω-categorical theories and theories with a unique countable recursively saturated model

Year: 2005
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