Historic forcing for Depth (Q2717744)

The depth of a Boolean algebra \({\mathfrak B}\), \(\text{Depth}({\mathfrak B}\)), is the supremum of all cardinalities of well-ordered subsets \({\mathfrak B}\). NEWLINENEWLINENEWLINEIn this paper the authors show consistently that for regular cardinals \(\theta, \lambda\) with \(\theta <\lambda\) there exists a Boolean algebra \({\mathfrak B}\) with \(|{\mathfrak B}|=\lambda^+\) such that for every subalgebra \({\mathfrak B}'\subset {\mathfrak B}\) with \(|{\mathfrak B}'|= \lambda^+\), we have \(\text{Depth} ({\mathfrak B}') =\theta\). This partially answers a question of Monk. NEWLINENEWLINENEWLINEThe result is obtained by means of a special forcing, the so-called historic forcing, which has been developed by Shelah some years ago and has been used by him already to obtain consistency results in partition calculus.