  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.
