An ideal game

• F. Galvin

• T. Jech

• M. Magidor

抄録

Let us consider the following infinite game between two players, Empty and Nonempty. We are given a large set S . Empty opens the game by choosing a large subset S 0 of S ; then Nonempty chooses a large set S 1 ⊆ S 0 ; then Empty chooses large S 2 ⊆ S , etc. The game is over after ω moves. If ⋂<jats:sub arrange="stack">n=0 <jats:sup arrange="stack">x S n is empty then Empty wins, and if ⋂<jats:sub arrange="stack">n=0 <jats:sup arrange="stack">∞ S n is nonempty then Nonempty wins. If “large” means “infinite”, then Empty can beat Nonempty rather easily: he chooses So countable, S 0 = { a 0 , a 1 ,…, a n ,…}, and then he chooses S 2 such that a 0 ∉ S 2 , S 4 such that a 1 , ∉ S 4 and so on. Next we assume that S is a set of uncountable cardinality, and that “large” means “of cardinality ∣ S ∣”. Then still Empty can win, but his winning strategy is somewhat more sophisticated: Let us identify S with a cardinal number κ. Thus each subset of S of size κ is a set of ordinals below κ. For each X ⊆ κ of size κ, let fx be the unique order-preserving mapping of X onto κ, and let F(X) = { x ϵ X : f(x) is a successor ordinal}. Empty's strategy is to play S 0 = F(K) , and when Nonempty plays S 2 k − 1 , let S 2 k = F(S 2k − 1 ).

収録刊行物

• Journal of Symbolic Logic

  Journal of Symbolic Logic 43 (2), 284-292, 1978-06

  Cambridge University Press (CUP)

キーワード

• Logic
• Philosophy

詳細情報

• CRID: 1362544419518629760
• NII論文ID: 30021529878
• DOI: 10.2307/2272827
• ISSN: 19435886
• ISSN: 00224812
• Web Site: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002248120004963X
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