Journal of Symbolic Logic

A Model in which Every Boolean Algebra has many Subalgebras

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

https://projecteuclid.org/euclid.jsl/1183744818

Mathematical Reviews number (MathSciNet)
MR1349006

Zentralblatt MATH identifier
0838.03038

JSTOR