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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 2 κ ≥ 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(More)