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A short proof of two recently discovered independence results using recursion theoretic methods


Author: E. A. Cichon
Journal: Proc. Amer. Math. Soc. 87 (1983), 704-706
MSC: Primary 03F30; Secondary 03D20, 10N15
MathSciNet review: 687646
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Abstract: Recently L. A. S. Kirby and J. Paris showed that a theorem of R. L. Goodstein cannot be proved in Peano's Arithmetic. We give an alternative short proof of their result, based only on well established results concerning recursion theoretic hierarchies of functions. A second, closely related result, due to F. S. Beckman and K. McAloon, is proved by the same means.


References [Enhancements On Off] (What's this?)

  • [1] F. S. Beckman and K. Mc Aloon, A direct proof of a result of Goodstein-Kirby-Paris, Lecture Notes, AMS Summer Institute on Recursion Theory, Cornell Univ., Ithaca, N. Y., June 28-July 16, 1982.
  • [2] Laurie Kirby and Jeff Paris, Accessible independence results for Peano arithmetic, Bull. London Math. Soc. 14 (1982), no. 4, 285–293. MR 663480, 10.1112/blms/14.4.285
  • [3] S. S. Wainer, A classification of the ordinal recursive functions, Arch. Math. Logik Grundlagenforsch. 13 (1970), 136–153. MR 0294134
  • [4] S. S. Wainer, Ordinal recursion, and a refinement of the extended Grzegorczyk hierarchy, J. Symbolic Logic 37 (1972), 281–292. MR 0321715

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0687646-0
Keywords: Goodstein's theorem, pure number base, cantor normal form, fundamental sequences, slow-growing hierarchy, Hardy hierarchy
Article copyright: © Copyright 1983 American Mathematical Society