5 - Games and the Silver property

Summary

Integer games connected with σ-ideals

Many of the σ-ideals considered in this book have integer games associated with them. As a result, they satisfy several interconnected properties, among them the selection property that will be instrumental in upgrading the canonization results to Silver-style dichotomies for these σ-ideals.

The subject of integer games and σ-ideals was treated in Zapletal (2008) rather extensively, but on a case-by-case basis. In this section, we provide a general framework, show that it is closely connected with uniformization theorems, and prove a couple of dichotomies under the assumption of the Axiom of Determinacy.

To help motivate the following definitions, we will consider a simple task. Let I be a collection of subsets of a Polish space X, closed under subsets. Suppose that I has a basis, a Borel set B ⊂ ω^ω × X such that a subset of X is in I iff it is covered by a vertical section of B. Let A ⊂ X be a set, and consider an infinite game in which Player I produces a point y ∈ ω^ω and Player II a point x ∈ X. Player II wins if x ∈ A \ B_y. Certainly, if A ∈ I then Player I has a winning strategy that completely disregards moves of Player II – just producing the vertical section of the basis which covers the set A.