### 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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