We continue our investigation on pcf with weak form of choice.
Characteristically we assume DC + P(Y) when looking and prod_{s in Y} delta_s.
We get more parallel of theorems on pcf.

We consider the random graph M^n_{\bar{p}} on the set [n], were the
probability of {x,y} being an edge is p_{|x-y|}, and \bar{p}=(p_1,p_2,p_3,...)
is a series of probabilities. We consider the set of all \bar{q} derived from
\bar{p} by inserting 0 probabilities to \bar{p}, or alternatively by decreasing
some of the p_i. We say that \bar{p} hereditarily satisfies the 0-1 law if the
0-1 law (for first order logic) holds in M^n_{\bar{q}} for any \bar{q} derived
from \bar{p} in the relevant way described above.

We deal with relatives of GCH which are provable. In particular we deal with
rank version of the revised GCH. Our motivation was to find such results when
only weak versions of the axiom of choice are assumed but some of the results
gives us additional information even in ZFC.

We prove a strong dichotomy for the number of ultrapowers of a given
countable model associated with nonprincipal ultrafilters on N. They are either
all isomorphic, or else there are $2^{2^{\aleph_0}}$ many nonisomorphic
ultrapowers. We prove the analogous result for metric structures, including
C*-algebras and II$_1$ factors, as well as their relative commutants and
include several applications. We also show that the C*-algebra B(H) always has
nonisomorphic relative commutants in its ultrapowers associated with
nonprincipal ultrafilters on N.