Strong cardinals in the core model Original Research Article

We work with Steelʹs 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 View the MathML source for k measurable in View the MathML source, 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.