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 λ where λ is not the successor of a cardinal of cofinality at most κ. If cf > ω then it follows that o λ, and if cf = ωthen either o λ or {α: K o 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 ω1 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. (shrink)

We shall construct a smooth category of mice and embeddings in the core model for measures of order 0. The existence of such a category implies that the global principle □ holds in K. We then prove a much stronger, the so-called condensation-coherent version of global □. The key tool of the whole construction is a new criterion on preserving soundness under condensation.

We show that form, n≧1 the existence of a∏ n m indescribable cardinal is equiconsistent with the failure of the combinatorial principle at a∏ n m indescribable cardinal κ together with the Generalized Continuum Hypothesis.

We prove the following theorem: Suppose that there is a singular $\kappa$ with the set of $\alpha$ 's with $o(\alpha)=\alpha^{+n}$ unbounded in it for every $n < \omega$ . Then in a generic extesion there are two precovering sets which disagree about common indiscernibles unboundedly often.