A Note on the Martin-harrington Theorem

In This document, we give a proof of the Martin-Harrington theorem, which says that for a real x, x] exists if and only all Π1(x) games are determined. The presentation follows that in Martin’s Determinacy monograph, and is as self-contained as possible. We do, however assume some knowledge of determinacy and the theory of sharps for sets of ordinals. 1. From Sharps to Determinacy In this section, we prove Martin’s theorem that says that, for a given real x, if x exists, then all Π1 games are determined. The idea here is to take the proof of analytic determinacy from a measurable and tighten it up. We say that D ⊆ ω generates an open set A ⊆ ω if A = {x ∈ ω : (∃d ∈ D)(d ⊆ x)}. Lemma 1.1. Let T be a game tree, and A ⊆ [T ] such that there is a D ⊆ ω that generates either A or ¬A. Let M be a transitive class model of ZFC, and ≺ a well-ordering of field(T ) such that T,D,≺ are in M . There is a strategy σ ∈ M that is definable in M from D, and is a winning strategy for some player in the game G(A;T ). Furthermore, M |= “σ is a winning strategy for G({x ∈ ω : (∃d ∈ D)(d ⊆ x)}, ω)”. Proof. We will assume that D generates A. It will be clear that the proof can be adapted to the case that D generates ¬A. We will inductively define, for each ordinal α, a set Pα of even length positions in T . Let p ∈ P0 if and only if (∃d ∈ D)(d ⊆ p). For α > 0, let p ∈ Pα if and only if p ∈ P0, or there is a legal move q at p such that either (i) q ∈ D ∩ [T ], or (ii) q / ∈ [T ] and, for every legal move r at q, r ∈ ⋃ β<α Pβ . Note that this definition is absolute for M , that is for every α ∈ M ∩ ON, Pα is defined in M in the same manner that it is defined in V . Furthermore, note that the sequence is monotonically increasing in the sense that β < α =⇒ Pβ ⊆ Pα. Additionally, if there is some γ such that Pγ = Pγ+1, the sequence is in fact constant beyond γ. Since each Pα ⊆ T , and T ∈M , comprehension forM gives us that ⋃ α∈M∩ON Pα ∈ M . We may then define a function that maps p ∈ T of even length to the least α ∈ M ∩ ON such that p ∈ Pα+1 \ Pα. This function has a Σ1 definition, and thus by Σ1 replacement we see that, since M thinks the image of this function is 1