2019/07/08 by Juan P. Aguilera, Sandra Müller, Aguilera, Juan P. +1
Computer Science · Mathematics · #03E15 #03E45 #03E55 #03E60 #Advanced Topology and Set Theory #Artificial Intelligence in Games #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.1907.03583
openalex publication_date 2019/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M^\sharpn(ℝ) denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn(ℝ) the class-sized model obtained by iterating the topmost measure of Mn(ℝ) class-many times. We characterize the sets of reals which are Σ1-definable from ℝ over Mn(ℝ), under the assumption that projective games on reals are determined: (1) for even n, Σ1Mn(ℝ) = \Game^ℝΠ1n+1; (2) for odd n, Σ1Mn(ℝ) = \Game^ℝΣ1n+1. This generalizes a theorem of Martin and Steel for L(ℝ), i.e., the case n=0. As consequences of the proof, we see that determinacy of all projective games with moves in ℝ is equivalent to the statement that M^\sharpn(ℝ) exists for all n∈ℕ, and that determinacy of all projective games of length ω2 with moves in ℕ is equivalent to the statement that M^\sharpn(ℝ) exists and satisfies AD for all n∈ℕ.