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 According to [6], another proof appeared in an unpublished paper by Jäger.
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Afshari, B., Rathjen, M. (2012). Ordinal Analysis and the Infinite Ramsey Theorem. In: Cooper, S.B., Dawar, A., Löwe, B. (eds) How the World Computes. CiE 2012. Lecture Notes in Computer Science, vol 7318. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30870-3_1
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DOI: https://doi.org/10.1007/978-3-642-30870-3_1
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