Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem

We prove the following theorem: Given a ⊆ ω and 1 ≤ α < ωCK 1 , if for some η < א1 and all u ∈ WO of length η, a is Σα(u), then a is Σα. We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ1-Turing-determinacy implies the existence of 0#. A major step in delineating the precise connections between large cardinals and game-determinacy hypotheses is the well-known theorem: For any real a, Σ1(a) games are determined if and only if a # exists. The “if” part is due to D. A. Martin [Mr2], and the “only if” part is Leo Harrington’s [Hg] (1). Harrington’s proof of this result is quite complex, relying on a fine analysis due to John Steel [Sl] of the ordinal-collapse forcing relation (a variant of this proof is given in Mansfield and Weitkamp’s [MW].) We propose here a new, forcing-free and quite elementary proof, Theorem 3.9. Our proof is built upon a new ordinal-definability theorem, for reals, which is interesting in its own right, namely Theorem 2.4: For α < ωCK 1 , if a real is Σ 0 α in (all codes of ) some countable ordinal , it is Σα. A further simplification is brought about by the use of an easily defined game (Definition 3.2) avoiding metamathematical notions. In §4, using the same techniques, we sketch a proof of a related result of Harrington. I wish to thank Alain Louveau for inspiring conversations during early stages of this work. 1991 Mathematics Subject Classification: 03D55, 03D60, 03E15, 03E55, 03E60, 04A15. (1) For an excellent mathematical and chronological account of the context of this last result, describing inter alia the important contributions of H. Friedman and D. A. Martin, see Kanamori’s [Kn], §31.