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 $κ$ such that $2^κ> kappa^+$ then there is an inner model with a cardinal $κ$ such that for all ordinals $α<κ$ there is an ordinal $ν< κ$ with $o(ν) > α$. (ii) If there is a singular strong limit cardinal $κ$ of uncountable cofinality such that $2^κ> κ^+$ then there is an inner model with $o(κ) = κ^{++}$. Since this paper was originally submitted, Gitik has improved this result to give exact lower bounds.