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Boole vs Wadge: Comparing Two Basic Tools of Descriptive Set Theory

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Revolutions and Revelations in Computability (CiE 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13359))

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Abstract

We systematically compare \(\omega \)-Boolean classes and Wadge classes, e.g. we complement the result of W. Wadge that the collection of non-self-dual levels of his hierarchy coincides with the collection of classes generated by Borel \(\omega \)-ary Boolean operations from the open sets in the Baire space. Namely, we characterize the operations, which generate any given level in this way, in terms of the Wadge hierarchy in the Scott domain. As a corollary we deduce the non-collapse of the latter hierarchy. Also, the effective version of this topic is developed.

This work was supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2022-281 from 05.04.2022 with the Ministry of Science and Higher Education of the Russian Federation.

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Authors and Affiliations

  1. A.P. Ershov Institute of Informatics Systems, and S.L. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Victor Selivanov

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  1. Victor Selivanov
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Correspondence to Victor Selivanov .

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Editors and Affiliations

  1. Department of Computer Science, Swansea University, Swansea, UK

    Ulrich Berger

  2. Department of Mathematics, Hofstra University, Hempstead, NY, USA

    Johanna N. Y. Franklin

  3. Georg-August University Göttingen, Göttingen, Germany

    Florin Manea

  4. Department of Computer Science, Swansea University, Swansea, UK

    Arno Pauly

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Selivanov, V. (2022). Boole vs Wadge: Comparing Two Basic Tools of Descriptive Set Theory. In: Berger, U., Franklin, J.N.Y., Manea, F., Pauly, A. (eds) Revolutions and Revelations in Computability. CiE 2022. Lecture Notes in Computer Science, vol 13359. Springer, Cham. https://doi.org/10.1007/978-3-031-08740-0_24

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  • DOI: https://doi.org/10.1007/978-3-031-08740-0_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-08739-4

  • Online ISBN: 978-3-031-08740-0

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