Journal of Symbolic Logic

Power-collapsing games

Miloš S. Kurilić and Boris Šobot

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Abstract

The game 𝔖ls (κ) is played on a complete Boolean algebra 𝔹, by two players, White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p∈𝔹. In the α-th move White chooses pα ∈(0,p)𝔹 and Black responds choosing iα ∈{0,1}. White wins the play iff \bigwedgeβ ∈κ α ≥ β pα iα =0, where pα ⁰=pα and pα ¹=p∖ pα . The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π (𝔹) or if 𝔹 contains a κ⁺-closed dense subset. On the other hand, if White has a w.s., then κ ∈ [𝔥₂(𝔹), π(𝔹)). The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2 < κ =κ ∈ Reg and forcing by 𝔹 preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S⊆ Reg there is a c.B.a. 𝔹 such that White (respectively, Black) has a w.s. for each infinite cardinal κ∈ S (resp. κ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game 𝔖ls(κ) is undetermined.

Article information

Source
J. Symbolic Logic, Volume 73, Issue 4 (2008), 1433-1457.

Dates
First available in Project Euclid: 27 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1230396930

Digital Object Identifier
doi:10.2178/jsl/1230396930

Mathematical Reviews number (MathSciNet)
MR2467228

Zentralblatt MATH identifier
1159.03035

Subjects
Primary: 91A44: Games involving topology or set theory 03E40: Other aspects of forcing and Boolean-valued models 03E35: Consistency and independence results 03E05: Other combinatorial set theory 03G05: Boolean algebras [See also 06Exx] 06E10: Chain conditions, complete algebras

Keywords
Boolean algebras games Suslin trees forcing

Citation

Kurilić, Miloš S.; Šobot, Boris. Power-collapsing games. J. Symbolic Logic 73 (2008), no. 4, 1433--1457. doi:10.2178/jsl/1230396930. https://projecteuclid.org/euclid.jsl/1230396930


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