Abstract
We start by giving a survey to the theory of \({\text {Borel}}^{*}(\kappa )\) sets in the generalized Baire space \({\text {Baire}}(\kappa )=\kappa ^{\kappa }\). In particular we look at the relation of this complexity class to other complexity classes which we denote by \({\text {Borel}}(\kappa )\), \({\Delta _1^1}(\kappa )\) and \({\Sigma _1^1}(\kappa )\) and the connections between \({\text {Borel}}^*(\kappa )\) sets and the infinitely deep language \(M_{\kappa ^+\kappa }\). In the end of the paper we will prove the consistency of \({\text {Borel}}^{*}(\kappa )\ne \Sigma ^{1}_{1}(\kappa )\).
Partially supported by the Academy of Finland through its grant WBS 1251557.
Research supported by the Science Foundation of the University of Helsinki.
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Hyttinen, T., Kulikov, V. (2018). Borel\(^{*}\) Sets in the Generalized Baire Space and Infinitary Languages. In: van Ditmarsch, H., Sandu, G. (eds) Jaakko Hintikka on Knowledge and Game-Theoretical Semantics. Outstanding Contributions to Logic, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-62864-6_16
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