The canonical function game

The canonical function game is a game of length ω1 introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ2 absoluteness, cardinality spectra and Π2 maximality for H(ω2) relative to the Continuum Hypothesis. MSC2000: 03E60; 03E50, 03D60 The canonical function game, introduced by W.H. Woodin, is a game of perfect information of length ω1 between two players, whom we call Dominating and Undominated. In each round α, Undominated plays a countable ordinal u(α), and then Dominating plays σα, a wellordering of α of ordertype greater than u(α) (if α ≥ ω; when α is finite we require only that σα is a wellordering of α; the first ω moves are irrelevant to the outcome of the game). After all ω1 rounds have been played, Dominating wins the run of the game if and only if there exists a club C ⊂ ω1 such that σα = σβ ∩ (α× α) for all α < β in C. Given an ordinal γ ∈ [ω1, ω2), a canonical function for γ is a function f : ω1 → ω1 for which there exists a bijection π : ω1 → γ such that the set {α < ω1 | f(α) = o.t.(π[α])} contains a club subset of ω1. Any two canonical functions for the same ordinal agree on a club. Futhermore, if γ < γ′ are ordinals in [ω1, ω2), f is a canonical function for γ and f ′ is a canonical function for γ′, then f ′ > f on a club. If 〈(u(α), σα) : α < ω1〉 is a run of the canonical function game and C ⊂ ω1 is a club witnessing that Dominating wins this run of the game, then Σ = ∪{σα : α ∈ C} is a wellordering of ω1, and the function f : ω1 → ω1 defined by letting f(α) be the ordertype of σα is a canonical function for the ordertype of Σ. ∗The research in this paper was conducted while the author was a guest of the Fields Institute, and the writing was completed with the support of a FAPESP fellowship (Grant # 02/11551-3) at the University of São Paulo.