Splitting Number and the Core Model

We provide a lower bound for the consistency strength of the hypothesis proposed by S. Kamo: ∃κ > א0 s(κ) ≥ κ. 0. Introduction. In [3] ,[10] cardinal invariants on ω are generalized for uncountable regular cardinals and it is shown that some of their properties still hold true for their uncountable counterparts.However,the role of the splitting number s(κ) (for the definition see below) changes significantly. As Suzuki [10] observed,existence of κ > א0 regular such that s(κ) ≥ κ implies existence of inner models with measurable cardinals and later Kamo [5] proved that ∃κ > א0 s(κ) ≥ κ ++ is consistent provided there is 2-supercompact cardinal κ.Here we give a lower bound for the consistency strength of the statement ∃κ > א0 s(κ) ≥ κ . Theorem. Cons ( ∃κ > א0 s(κ) ≥ κ ++ ) → Cons ( ∃α o(α) ≥ α ).