Using a game characterization of distributivity, we show that base matrices for P(ω)/fin of regular height larger than h necessarily have maximal branches which are not cofinal.
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 $${\mathcal {B}}$$
B
-Canjar for any countably directed unbounded family $${\mathcal {B}}$$
B
of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that $${\mathfrak {b}}=\omega _1$$
b
=
ω
1
in every extension by the above forcing notions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.