On the Singular Cardinal Hypothesis

Abstract

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.