TeXipedia

Game

Represents a game quantifier indicating the existence of a winning strategy for Player 1 in two-player games and logical systems.

Overview

Essential in mathematical logic, game theory, and descriptive set theory for formalizing strategic interactions and determinacy statements.

  • Commonly used in game-theoretic semantics to express alternating quantification
  • Appears in formal proofs involving infinite games and determinacy theorems
  • Helps formalize statements about winning strategies in two-player perfect information games
  • Particularly important in set theory when discussing determinacy axioms and their consequences

Examples

Expressing the existence of a winning strategy in a game-theoretic formula.

⅁x∃y∀zP(x,y,z)\Game x \exists y \forall z P(x,y,z) 
\Game x \exists y \forall z P(x,y,z)

Game quantifier in a logical statement about determinacy.

⅁x∈X  (ϕ(x)∨ψ(x))\Game x \in X \; (\phi(x) \lor \psi(x)) 
\Game x \in X \; (\phi(x) \lor \psi(x))

Using game quantifier in set-theoretic context.

⅁x∈ωω∃n∈ω(x(n)=0)\Game x \in \omega^\omega \exists n \in \omega (x(n) = 0) 
\Game x \in \omega^\omega \exists n \in \omega (x(n) = 0)