Countable Ordinals and the Analytical Hierarchy, I

Proof. For notational simplicity let us take n = 1 as a typical case. Thus let A C WO be ΣJ and assume sup {|a |: a E A} < Nj. Let B Qω be Π2 and /: ω ->ω recursive such that f[B] = A. Consider then the following game: Player I plays β, player II plays γ and II wins iff γ E WO&(β E B -> \f(β)| g | γ | ) . Clearly player II has a winning strategy in this game. But his payoff set is Σ\9 so by a result of Moschovakis [6] he has a winning strategy r which is ΔJ. Let T = {/3 * r : jβ E ω}, where β * τ is the result of IΓs moves following r when I plays β. Then Γ C WO and Γ is Σ|(τ), so by the Boundedness Theorem