Winning strategies in club games and their applications

摘要

We present results concerning winning strategies and tactics in club games on Pω1λ. We show that there is generally no winning tactic for the player trying to get inside the club. The bound-countable game turns out to be rather fruitful and adds to some previous results about the construction of elementary substructures and their localization in certain intervals. We show that Player II has a winning strategy in the bound-countable game, thus establishing a new ZFC result. The applications given are new proofs for two cardinal diamonds and the impossibility of collapsing cardinals to 2 under certain conditions.