Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Background facts
- 3 Analytic equivalence relations and models of set theory
- 4 Classes of equivalence relations
- 5 Games and the Silver property
- 6 The game ideals
- 7 Benchmark equivalence relations
- 8 Ramsey-type ideals
- 9 Product-type ideals
- 10 The countable support iteration ideals
- References
- Index
5 - Games and the Silver property
Published online by Cambridge University Press: 18 December 2013
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Background facts
- 3 Analytic equivalence relations and models of set theory
- 4 Classes of equivalence relations
- 5 Games and the Silver property
- 6 The game ideals
- 7 Benchmark equivalence relations
- 8 Ramsey-type ideals
- 9 Product-type ideals
- 10 The countable support iteration ideals
- References
- Index
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 ⊂ ωω × 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 ∈ ωω and Player II a point x ∈ X. Player II wins if x ∈ A \ By. 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.
- Type
- Chapter
- Information
- Canonical Ramsey Theory on Polish Spaces , pp. 80 - 99Publisher: Cambridge University PressPrint publication year: 2013