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A model in which every Boolean algebra has many subalgebras

Published online by Cambridge University Press:  12 March 2014

James Cummings
Mathematics Institute, Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel, E-mail:
Saharon Shelah
Mathematics Institute, Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel, E-mail:


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 2A = 2B. This implies in particular that B has 2B 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.

Research Article
Copyright © Association for Symbolic Logic 1995

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