  We use the core model for sequences of measures to prove a new lower bound
for the consistency strength of the failure of the SCH:
  THEOREM
  (i) If there is a singular strong limit cardinal $\kappa$ such that $2^\kappa
> kappa^+$ then there is an inner model with a cardinal $\kappa$ such that for
all ordinals $\alpha<\kappa$ there is an ordinal $\nu < \kappa$ with $o(\nu) >
\alpha$.
  (ii) If there is a singular strong limit cardinal $\kappa$ of uncountable
cofinality such that $2^\kappa > \kappa^+$ then there is an inner model with
$o(\kappa) = \kappa^{++}$.
  Since this paper was originally submitted, Gitik has improved this result to
give exact lower bounds.
