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By William J. Mitchell


Abstract. We reprove Gitik’s theorem that if the GCH holds and o(κ) = κ + 1 then there is a generic extension in which κ is still measurable and there is a closed unbounded subset C of κ such that every ν ∈ C is inaccessible in the ground model. Unlike the forcing used by Gitik, the iterated forcing Rλ+1 used in this paper has the property that if λ is a cardinal less then κ then Rλ+1 can be factored in V as Rκ+1 = Rλ+1 × Rλ+1,κ where |Rλ+1 | ≤ λ + and Rλ+1,κ does not add any new subsets of λ. §1. Introduction. In [2], Gitik proves the following theorem: Theorem 1.1 (Gitik). If V satisfies the GCH and o(κ)> κ then there is a generic extension V [ � C] in which κ is still measurable, but in which there is a closed, unbounded subset Cκ of κ such that every member of Cκ is inaccessible in V

Year: 2009
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