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Ordinal Analysis and the Infinite Ramsey Theorem

  • Bahareh Afshari
  • Michael Rathjen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)

Abstract

The infinite Ramsey theorem is known to be equivalent to the statement ‘for every set X and natural number n, the n-th Turing jump of X exists’, over RCA0 due to results of Jockusch [5]. By subjecting the theory RCA0 augmented by the latter statement to an ordinal analysis, we give a direct proof of the fact that the infinite Ramsey theorem has proof-theoretic strength ε ω . The upper bound is obtained by means of cut elimination and the lower bound by extending the standard well-ordering proofs for ACA0. There is a proof of this result due to McAloon [6], using model-theoretic and combinatorial techniques. According to [6], another proof appeared in an unpublished paper by Jäger.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bahareh Afshari
    • 1
  • Michael Rathjen
    • 2
  1. 1.School of InformaticsUniversity of EdinburghEdinburghUnited Kingdom
  2. 2.School of MathematicsUniversity of LeedsLeedsUnited Kingdom

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