Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis

Abstract

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 $\cofinality(κ)>\gw$ then $o(\kappa )\ge \lambda$. (ii) If $\cofinality(κ)=\gw$ 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.

References in corpus (1)

• On hidden extenders

Cited by in corpus (2)

• Some results on nonstationry ideal 2
• Blowing up the power of a singular cardinal