A game on Boolean algebras describing the collapse of the continuum

Abstract

The game G ls is played on a complete Boolean algebra B in ω -many moves. At the beginning White chooses a non-zero element p of B and, in the n th move, White chooses a positive p n < p and Black responds by choosing an i n ∈ { 0 , 1 } . White wins the play iff lim sup p n i n = 0 . It is shown that White has a winning strategy in this game iff forcing by B collapses the continuum to ω in some generic extension. On the other hand, if a complete Boolean algebra B carries a strictly positive Maharam submeasure or contains a countable dense subset, then Black has a winning strategy in the game G ls played on B . A Suslin algebra on which the game is undetermined is constructed and the game G ls is compared with the well-known cut-and-choose games G c&c , G fin ( λ ) and G ω ( λ ) introduced by Jech.