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A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with o(κ) = κ + 1. On the other hand, it is shown that the strength of above is at least o(κ) = κ.

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)