Strong Cardinals in the Core Model

We work with Steel's core model under the assumption that there is no inner class model for a Woodin cardinal. If there is no ω1V strong cardinal in the Steel core model, then K ∩ HC is projective. Moreover, if V = MColl(ω, < κ) for k measurable in M, then K is projective up to the first < ω1V-strong. This is used to resolve negatively the boldface correctness conjecture from Hauser (1995). We also show in ZFC that set forcing cannot create class models with a given number of strongs.

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