Abstract

It is proved that the following conditions are equivalent: (a) there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra, (b) there exists a nowhere dense ultrafilter on ω. Therefore, the existence of such algebras is undecidable in ZFC. In "forcing language" condition (a) says that there exists a non-trivial σ-centered forcing not adding Cohen reals