Archive for Mathematical Logic

, Volume 47, Issue 6, pp 529–548

Register computations on ordinals


DOI: 10.1007/s00153-008-0093-3

Cite this article as:
Koepke, P. & Siders, R. Arch. Math. Logic (2008) 47: 529. doi:10.1007/s00153-008-0093-3


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.


Ordinal computability Hypercomputation Infinitary computation Register machine 

Mathematics Subject Classification (2000)

03D60 03E45 

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematisches Institut, University of BonnBonnGermany
  2. 2.Department of MathematicsUniversity of HelsinkiHelsinkiFinland