Propagation of the scale property using games
Itay Neeman · Cambridge University Press eBooks · 2008
The aim of this short paper is to introduce the reader to the notion of a scale and to some of the basic techniques involved in the propagation of the scale property through the use of infinite games. None of the results presented is due to the author. For a full history see Moschovakis [2]. We work throughout the paper with the space ω ω. For s ∈ ω <ω we use Ns to denote the set {x ∈ ω ω | x extends s}. The sets Ns, s ∈ ω <ω, form the basic open subsets of ω ω. Following standard abuse we refer to the space ω ω, equipped with the topology generated by these basic open set, as R. Given a set A ⊂ R let G(A) denote the following game: Players I and II alternate playing x(n) for n ∈ ω subject to the order displayed in Diagram 1, with x(n) ∈ ω for each n. If, after ω moves, the real x = 〈x(n) | n < ω 〉 belongs to A then player I wins. Otherwise player II wins. G(A) is determined if one of the two players has a winning strategy in the game. I x(0) x(2)...... II x(1) x(3)...... Diagram 1. The game G(A). For B ⊂ R × R and x ∈ R let Bx = {y ∈ R | 〈x,y 〉 ∈ B}. This is the x–section of B. Define �B to be the set {x ∈ R | player I has a winning strategy in G(Bx)}. We sometimes write (�y)B(x,y), or (�y)〈x,y 〉 ∈ B, for the statement x ∈ �B. This is deliberately meant to conjure up the notation used for statements involving the quantifiers (∀y) and (∃y). (�y) really is a quantifier, giving precise meaning to the chain (∃y(0))(∀y(1))(∃(y(2)) · · · · · · of quantifiers over ω. Let B ⊂ R×R be open. Note that for each 〈x,y 〉 ∈ B there exists some n < ω so that Nx↾n ×Ny↾n ⊂ B. Let n(x,y) denote the least such n. We refer to n(x,y) as the time of entry of 〈x,y 〉 into B. For 〈x,y 〉 ∈ B we set n(x,y) = ω. Let A = �B. For x,x ∗ ∈ R define H(x ∗,x) to be the following game: Players “first ” and “second ” (denoted F and S respectively) alternate moves subject to the format in Diagram 2. The moves are played sequentially from left to right, and are presented in two separate lines only for future convenience. The letters F and S indicate which player is responsible for each move. Each of the moves is a natural number. An infinite run leading to reals y = 〈y(i) | i < ω 〉 and