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.