A model in which every infinite Boolean algebra has many subalgebras

by Cummings and Shelah. [CuSh:530]
J Symbolic Logic, 1995
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. 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.

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