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 $2^{\kappa }>kapp{a}^{+}$ 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.