Power-collapsing games. (English) Zbl 1159.03035

Kurilić, Miloš S.; Šobot, Boris

J. Symb. Log. 73, No. 4, 1433-1457 (2008).

Authors’ abstract: “The game \({\mathcal G}_{\text{ls}}(\kappa)\) is played on a complete Boolean algebra \({\mathbb B}\) by two players, White and Black, in \(\kappa\)-many moves (where \(\kappa\) is an infinite cardinal). At the beginning White chooses a non-zero element \(p\in{\mathbb B}\). In the \(\alpha\)-th move, White chooses \(p_\alpha\in(0,p)_{\mathbb B}\) and Black responds by choosing \(i_\alpha\in\{0,1\}\). White wins the play iff \(\bigwedge_{\beta\in\kappa}\bigvee_{\alpha\geq\beta}p_\alpha^{i_\alpha}=0\), where \(p^0_\alpha=p_\alpha\) and \(p^1_\alpha=p\backslash p_\alpha\).
The corresponding game-theoretic properties of c.B.a.’s are investigated. So, Black has a winning strategy (w.s.) if \(\kappa\geq\pi({\mathbb B})\) or if \({\mathbb B}\) contains a \(\kappa^+\)-closed dense subset. On the other hand, if White has a w.s., then \(\kappa\in[{\mathfrak h}_2({\mathbb B}),\pi({\mathbb B}))\). The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if \(2^{<\kappa}=\kappa\in\text{Reg}\) and forcing by \({\mathbb B}\) preserves the regularity of \(\kappa\), then White has a w.s. iff the power \(2^\kappa\) is collapsed to \(\kappa\) in some extension. It is shown that under the GCH, for each set \(S\subseteq\text{Reg}\) there is a c.B.a. \({\mathbb B}\) such that White (respectively Black) has a w.s. for each infinite cardinal \(\kappa\in S\) (resp. \(\kappa\notin S\)). Also it is shown consistent that for each \(\kappa\in\text{Reg}\) there is a c.B.a. on which the game \({\mathcal G}_{\text{ls}}(\kappa)\) is undetermined.”

Reviewer: James Monk (Boulder)

MSC:

03E40	Other aspects of forcing and Boolean-valued models
06E10	Chain conditions, complete algebras
91A44	Games involving topology, set theory, or logic

Keywords:

Boolean algebras; games; Suslin trees; forcing

References:

[1]	Set theory, An introduction to independence proofs (1980) · Zbl 0443.03021
[2]	Set theory (1997)
[3]	DOI: 10.1090/S0002-9939-02-06501-2 · Zbl 1004.03040 · doi:10.1090/S0002-9939-02-06501-2
[4]	DOI: 10.1090/S0002-9939-03-07197-1 · Zbl 1013.03501 · doi:10.1090/S0002-9939-03-07197-1
[5]	DOI: 10.1016/0168-0072(95)00002-X · Zbl 0849.03043 · doi:10.1016/0168-0072(95)00002-X
[6]	DOI: 10.1016/0168-0072(84)90038-1 · Zbl 0555.03025 · doi:10.1016/0168-0072(84)90038-1