A forcing construction of thin-tall Boolean algebras

Juan Carlos Martínez¹

¹

Fundamenta Mathematicae, Tome 159 (1999) no. 2, pp. 99-113

Résumé

It was proved by Juhász and Weiss that for every ordinal α with ${0 α ω_2}$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that $κ^{ κ} = κ$ and α is an ordinal such that $0 α κ^{++}$, then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating this forcing through all $α κ^{++}$, we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic extension there is a superatomic Boolean algebra of height α and width κ for every $α κ^{++}$. Consistency for specific κ, like $ω_1$, then follows as a corollary.

Détail

DOI : 10.4064/fm-159-2-99-113