Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-29T02:24:33.860Z Has data issue: false hasContentIssue false

Concerning n-tactics in the countable-finite game

Published online by Cambridge University Press:  12 March 2014

Marion Scheepers*
Affiliation:
Department of Mathematics, Boise State University, Boise, Idaho 83725

Extract

In the paper [S1] I introduced a game, denoted by MG(J) (where J is a free ideal on some infinite set S) and called “the meager nowhere dense game for J”. The special case when J is the collection of finite subsets of the set S is called the countable-finite game on S. It proceeds as follows.

First player ONE picks a countable set C 1, then player TWO picks a finite set F 1. Then in the second inning ONE picks a countable set C 2 with C 1C 2 (unless explicitly indicated otherwise, “⊂” means “is a proper subset of”) and TWO responds with a finite set F 2, and so on. The players construct a sequence (C1,F1,C2,F2,…,Ck,Fk,…) where for each positive integer k

(i) Ck denotes ONE's countable set picked during the kth inning,

(ii) Fk denotes TWO's finite set picked during the kth inning, and

(iii) Ck ⊂ Ck + 1.

Such a sequence is a play of the countable-finite game on S, and TWO wins this play if is contained in . The notion of a winning perfect information strategy is defined as usual (see, for example, [S1]). Zermelo-Fraenkel set theory together with the axiom of choice (denoted by ZFC; for a statement of the axioms see pp. xv–xvi of [K]) is a strong enough theory to build a winning perfect information strategy for player TWO in this game.

Does TWO have a winning strategy requiring less than perfect information? Fix a positive integer k. A strategy of TWO which requires knowledge of only at the most the k most recent moves of ONE is said to be a k-tactic. For the countable-finite game on an infinite set S the following facts about the existence of winning k-tactics for TWO are proved in [S1]:

1) TWO does not have a winning 1-tactic (Theorem 1 of [S1]).

2) If the cardinality of S is less than ℵ2 then TWO has a winning 2-tactic (Corollary 4 of [S1]).

3) If TWO has a winning k-tactic in the countable-finite game on an infinite set S, then TWO has a winning 3-tactic (Proposition 15 of [SI]).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[K] Kunen, Kenneth, Set theory: an introduction to independence proofs, North-Holland, Amsterdam, 1980.Google Scholar
[L] Laver, Richard, Linear orders in (ω)ω under eventual dominance, Logic Colloquium '78, North-Holland, Amsterdam, 1979, pp. 299302.Google Scholar
[S1] Scheepers, Marion, Meager-nowhere dense games. I: n-tactics, Rocky Mountain Journal of Mathematics (to appear).Google Scholar
[S2] Scheepers, Marion, A partition relation for partially ordered sets, Order, vol. 7 (1990), pp. 4164.CrossRefGoogle Scholar
[Tl] Todorčević, Stevo, Partitioning pairs of countable ordinals, Acta Mathematica, vol. 159 (1987), pp. 261294.CrossRefGoogle Scholar
[T2] Todorčević, Stevo, Partition relations for partially ordered sets, Acta Mathematica, vol. 155 (1985), pp. 125.CrossRefGoogle Scholar