Journal of Symbolic Logic

A Model in which Every Boolean Algebra has many Subalgebras

James Cummings and Saharon Shelah

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Abstract

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

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

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744818

Mathematical Reviews number (MathSciNet)
MR1349006

Zentralblatt MATH identifier
0838.03038

JSTOR
links.jstor.org

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

Keywords
Boolean algebras free subsets Radin forcing

Citation

Cummings, James; Shelah, Saharon. A Model in which Every Boolean Algebra has many Subalgebras. J. Symbolic Logic 60 (1995), no. 3, 992--1004. https://projecteuclid.org/euclid.jsl/1183744818


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