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 <omega(1)(V) s trong cardinal in the Steel core model, then K boolean AND HC is proje ctive. Moreover, if V=M(Coll(omega,<kappa)) for kappa measurable in M, then K is projective up to the first <omega(1)(V)-strong. This is use d to resolve negatively the boldface correctness conjecture from Hause r (1995). We also show in ZFC that set forcing cannot create class mod els with a given number of strongs.