# Sh:530

- Cummings, J., & Shelah, S. (1995).
*A model in which every Boolean algebra has many subalgebras*. J. Symbolic Logic,**60**(3), 992–1004. arXiv: math/9509227 DOI: 10.2307/2275769 MR: 1349006 -
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. 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. - Current version: 1995-09-04_10 (22p) published version (14p)

Bib entry

@article{Sh:530, author = {Cummings, James and Shelah, Saharon}, title = {{A model in which every Boolean algebra has many subalgebras}}, journal = {J. Symbolic Logic}, fjournal = {The Journal of Symbolic Logic}, volume = {60}, number = {3}, year = {1995}, pages = {992--1004}, issn = {0022-4812}, mrnumber = {1349006}, mrclass = {03E35 (03G05)}, doi = {10.2307/2275769}, note = {\href{https://arxiv.org/abs/math/9509227}{arXiv: math/9509227}}, arxiv_number = {math/9509227} }