Projective Set (game)
In game theory, a projective set, within the context of descriptive set theory applied to games, refers to a set of real numbers that can be obtained from Borel sets through alternating projections and complementations. More formally, the projective hierarchy is a transfinite hierarchy of sets of real numbers, starting with Borel sets and extending through alternating operations of projection and complementation.
The projective sets are constructed inductively. The Borel sets form the base of the hierarchy, denoted as Σ01 = Π01. Subsequent levels are defined recursively. A set A is Σ1n+1 if it is the projection of a Π1n set, meaning there exists a Π1n set B in the product space X × ℝ (where X is some suitable Polish space, such as ℝ or the Cantor space) such that x ∈ A if and only if there exists a y ∈ ℝ such that ( x, y ) ∈ B. A set A is Π1n+1 if its complement is Σ1n+1. The projective sets are the union of all Σ1n and Π1n sets for all natural numbers n.
The study of projective sets is relevant to determinacy results in game theory. A key result connecting descriptive set theory and game theory is that if all projective sets are determined, then many pathological situations, such as the existence of non-measurable sets, can be avoided. The Axiom of Determinacy (AD), which asserts that all two-player perfect information games on natural numbers are determined, implies that all projective sets have desirable regularity properties, like being Lebesgue measurable, having the Baire property, and possessing the perfect set property. However, AD is incompatible with the Axiom of Choice (AC). The weaker axiom of projective determinacy (PD), which asserts that all projective games are determined, is consistent with ZFC (Zermelo–Fraenkel set theory with the Axiom of Choice) and provides a framework within which projective sets behave "nicely."