Forcing, games and families of closed sets

2009/10/13 by Marcin Sabok, Sabok, Marcin
Economics, Econometrics and Finance · Mathematics · #03E15 #28A05 #54H05 #Advanced Topology and Set Theory #Economic theories and models #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO #msc:03E15 #msc:28A05 #msc:54H05

paper · pdf · doi:10.48550/arxiv.0910.2318

arxiv created 2009/10/13 · openalex publication_date 2009/10/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. This generalizes the classical approach to the Sacks and the Miller forcing. For definable (\mathbfΠ¹₁ on \mathbfΣ¹₁) σ-ideals we show that if a σ-ideal is generated by closed sets, then it is generated by closed sets in all forcing extensions. We also prove an infinite-dimensional version of the Solecki dichotomy for analytic sets. Among examples, we investigate the σ-ideal \E generated by closed null sets and σ-ideals connected with not piecewise continuous functions.

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