Journal of Symbolic Logic

A Model in which Every Boolean Algebra has many Subalgebras

James Cummings and Saharon Shelah

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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.

Article information

J. Symbolic Logic, Volume 60, Issue 3 (1995), 992-1004.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03E35: Consistency and independence results
Secondary: 03G05: Boolean algebras [See also 06Exx]

Boolean algebras free subsets Radin forcing


Cummings, James; Shelah, Saharon. A Model in which Every Boolean Algebra has many Subalgebras. J. Symbolic Logic 60 (1995), no. 3, 992--1004.

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