Towers, mad families, and unboundedness

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 [Formula: see text] -Canjar for any countably directed unbounded family [Formula: see text] of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that [Formula: see text] in every extension by the above forcing notions.