Remark on the altitude of Boolean algebras (Q1109798)

The altitude of a BA A is the least infinite cardinal \(\lambda\) such that there is an ultrafilter F on A which is the union of a strictly increasing sequence of type \(\lambda\) of filters on A. This notion is closely related to the cofinality of A, studied by S. Koppelberg. It is a big open problem whether the altitude is always at most \(\omega_ 1\); it is easy to see that it is always at most \(2^{\omega}\). Here the author shows that it is consistent that CH fails and still every BA has altitude at most \(\omega_ 1\).