Ordinal Analysis and the Infinite Ramsey Theorem

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


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 According to [6], another proof appeared in an unpublished paper by Jäger.


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© 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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