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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Koepke, P., Siders, R. Register computations on ordinals. Arch. Math. Logic 47, 529–548 (2008). https://doi.org/10.1007/s00153-008-0093-3
- Ordinal computability
- Infinitary computation
- Register machine
Mathematics Subject Classification (2000)