Towers, mad families, and unboundedness

Abstract

We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are \({\mathcal {B}}\)-Canjar for any countably directed unbounded family \({\mathcal {B}}\) of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that \({\mathfrak {b}}=\omega _1\) in every extension by the above forcing notions.