Woodin for strong compactness cardinals

We give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor's identity crisis theorem for the first strongly compact cardinal.Comment: 20 pages, fixed proof of Theorem 4.1, minor corrections and addition

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