Determinacy and the Structure of L(R)

Let ω = {0, 1, 2, ... } be the set of natural numbers and R = ω^ω the set of all infinite sequences from ω, or for simplicity reals. To each set A ⊆ R we associate a two-person infinite game, in which players I and II alternatively play natural numbers I x(0) x(2) II x(1) x(3)...x(O), x(l), x(2), ... and if x is the real they eventually produce, then I wins iff x є A. The notion of a winning strategy for player I or II is defined in the usual way, and we call A determined if either player I or player II has a winning strategy in the above game. For a collection ⌈ of sets of reals let ⌈-DET be the statement that all sets A є ⌈ are determined. Finally AD (The Axiom of Determinacy) is the statement that all sets of reals are determined.