Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis

We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $\kappa$ is a singular strong limit cardinal and $2^\kappa >= \lambda$ where $\lambda$ is not the successor of a cardinal of cofinality at most $\kappa$. (i) If $\cofinality(\kappa)>\gw$ then $o(\kappa)\ge\lambda$. (ii) If $\cofinality(\kappa)=\gw$ then either $o(\kappa)\ge\lambda$ or \set{\ga:K\sat o(\ga)\ge\ga^{+n}} is cofinal in $\kappa$ for each $n\in\gw$. In order to prove this theorem we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders.