Indiscernible sequences for extenders, and the singular cardinal hypothesis

AbstractWe 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 ⩾ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf(κ) > ω then it follows that o(κ) ⩾ λ, and if cf(κ) = ωthen either o(κ) ⩾ λ or {α: K ⊨ o(α) ⩾ α+n} is confinal in κ for each n ϵ ω.We also prove several results which extend or are related to this result, notably Theorem. If 2ω < ℵω and 2ℵω > ℵω1 then 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