Ordinal Analysis and the Infinite Ramsey Theorem
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 . 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 , using model-theoretic and combinatorial techniques. According to , another proof appeared in an unpublished paper by Jäger.
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