  The following results are proved:
  (a) In a model obtained by adding aleph_2 Cohen reals, there is always a
c.c.c. complete Boolean algebra without the weak Freese-Nation property.
  (b) Modulo the consistency strength of a supercompact cardinal, the existence
of a c.c.c. complete Boolean algebras without the weak Freese-Nation property
consistent with GCH.
  (c) Under some consequences of the negation of 0^#, the weak Freese-Nation
property of (P(omega),subseteq) is equivalent to the weak Freese-Nation
property of any of C(kappa) or R(kappa) for uncountable kappa.
  (d) Modulo consistency of (aleph_{omega+1},aleph_omega)-->(aleph_1,aleph_0),
it is consistent with GCH that the assertion in (c) does not hold and also that
adding aleph_omega Cohen reals destroys the weak Freese-Nation property of
(P(omega),subseteq)
