×

Winning strategies in club games and their applications. (English) Zbl 1232.03032

Author’s abstract: “We present results concerning winning strategies and tactics in club games on \(\mathcal P_{\omega_1}\lambda \). 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 \(\aleph_{2}\) under certain conditions.”

MSC:

03E05 Other combinatorial set theory
03E10 Ordinal and cardinal numbers
91A44 Games involving topology, set theory, or logic
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] K. Devlin Constructibility (Springer-Verlag, 1984).
[2] Donder, Two cardinal versions of diamond, Israel J. Mathematics 83 pp 1– (1993) · Zbl 0798.03047 · doi:10.1007/BF02764635
[3] Foreman, A new Löwenheim-Skolem theorem, Trans. Amer. Math. Soc. 357 pp 1693– (2005) · Zbl 1082.03034 · doi:10.1090/S0002-9947-04-03445-2
[4] S. Shelah Non-structure Theory (Oxford University Press).
[5] S. Shelah Cardinal Arithmetic (Oxford University Press, 1994).
[6] S. Shelah Proper and Improper Forcing. Perspectives in Mathematical Logic (Springer-Verlag, 1998) · Zbl 0889.03041
[7] M. Shioya Diamonds on {\(\kappa\)} {\(\lambda\)} (preprint).
[8] Velickovic, Forcing axioms and stationary sets, Advances Math. 94 pp 256– (1992) · Zbl 0785.03031 · doi:10.1016/0001-8708(92)90038-M
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.