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Eventually infinite time Turing machine degrees: infinite time decidable reals

  • P.D. Welch (a1) (a2)
Abstract
Abstract

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

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[1]Barwise K.J., Admissible sets and structures, Perspectives in Mathematical Logic, Springer Verlag, Berlin, Heidelberg, 1975.
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[7]Welch P.D., The length of infinite time Turing degree computations, Bulletin of the London Mathematical Society, vol. 32 (2000), no. 2, pp. 129–136.
[8]Welch P.D., Minimality arguments in the infinite time Turing degrees, Sets and proofs, Proceedings of the ASL European Meeting 1997 (Cooper S.B. and Truss J.K., editors), London Mathematical Society Lecture Notes in Mathematics, vol. 258, Cambridge University Press, April 1999, pp. 425–436.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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