Forcing, games and families of closed sets

Details Forcing, games and families of closed sets Forcing, games and families of closed sets

2009-10-13

arxiv.org/abs/0910.2318v1

Mathematics

Logic

Scientific paper

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 ($\mathbf{\Pi}^1_1$ on $\mathbf{\Sigma}^1_1) $\sigma$-ideals we show that if a $\sigma$-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 $\sigma$-ideal $\E$ generated by closed null sets and $\sigma$-ideals connected with not piecewise continuous functions.

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