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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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References

  1. 1

    Bissell-Siders, R.: Ordinal computers. Eprint at: arXiv:math.LO/9804076 (1998)

  2. 2

    Cutland, N.J.: Computability: An Introduction to Recursive Function Theory. Perspectives in Mathematical Logic. Cambridge University Press, London (1980)

  3. 3

    Devlin K.: Constructibility. Perspectives in Mathematical Logic. Springer, Berlin (1984)

  4. 4

    Gödel K.: The Consistency of the Continuum Hypothesis. In: Annals of Mathematical Studies, vol. 3. Princeton University Press, Princeton (1940)

  5. 5

    Hamkins J.D., Lewis A.: Infinite time turing machines. J. Symb. Log. 65(2), 567–604 (2000)

  6. 6

    Jech, T.: Set Theory, The Third Millennium Edition. In: Springer Monographs in Mathematics. Springer, Heidelberg (2003)

  7. 7

    Koepke P.: Turing computations on ordinals. Bull. Symb. Log. 11(3), 377–397 (2005)

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Author information

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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Keywords

  • Ordinal computability
  • Hypercomputation
  • Infinitary computation
  • Register machine

Mathematics Subject Classification (2000)

  • 03D60
  • 03E45