Project Euclid

We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra $B$ has an irredundant subset $A$ such that $2^{|A|} = 2^{|B|}$. This implies in particular that $B$ has $2^{|B|}$ subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a "black box" at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere.