Indiscernible sequences for extenders, and the singular cardinal hypothesis Original Research Article

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 κ is a singular strong limit cardinal and 2κ ⩾ λwhere λ is not the successor of a cardinal of cofinality at most κ. Ifcf(κ) > ωthen it follows thato(κ) ⩾ λ, and ifcf(κ) = ωthen eithero(κ) ⩾ λor{α: K ⊨ o(α) ⩾ α+n}is confinal in κ for eachnϵω. We also prove several results which extend or are related to this result, notably Theorem. IfView the MathML sourceandView the MathML sourcethen there is a sharp for a model with a strong cardinal. In order to prove these theorems we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders.