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

Measurable cardinals and good Σ1(κ)\Sigma_1(\kappa)-wellorderings

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $\kappa$ with the property that the collection of all initial segments of the wellordering is definable by a $\Sigma_1$-formula with parameter $\kappa$. A short argument shows that the existence of a measurable cardinal $\delta$ implies that such wellorderings do not exist at $\delta$-inaccessible cardinals of cofinality not equal to $\delta$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that interferes with the existence of such wellorderings at uncountable cardinals and we generalize the above result to the minimal model containing two measurable cardinals.Comment: 14 page