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.
