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Register computations on ordinals

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Abstract

We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register computable if and only if it is an element of Gödel’s constructible universe L.

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Authors and Affiliations

  1. Mathematisches Institut, University of Bonn, Beringstraße 1, 53115, Bonn, Germany

    Peter Koepke

  2. Department of Mathematics, University of Helsinki, P.O. Box 4, Yliopistonkatu 5, 00014, Helsinki, Finland

    Ryan Siders

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  1. Peter Koepke
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  2. Ryan Siders
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Correspondence to Peter Koepke.

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Koepke, P., Siders, R. Register computations on ordinals. Arch. Math. Logic 47, 529–548 (2008). https://doi.org/10.1007/s00153-008-0093-3

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  • DOI: https://doi.org/10.1007/s00153-008-0093-3

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