UC Irvine

The main result of this dissertation is a covering theorem for the core model below a Woodin cardinal. More precisely, we work with Steel's core model $\K$ constructed in $V_\Omega$ where $\Omega$ is measurable. The theorem is in a similar spirit to theorems of Mitchell and Cox and roughly says that either $\K$ recognizes the singularity of an ordinal $\kappa$ or else $\kappa$ is measurable in $\K$.

The first chapter of the thesis builds up the technical theory we will work in. The premice we work with use Mitchell-Steel indexing, but we use Jensen's $\Sigma^*$ fine structure and a different amenable coding. The use of $\Sigma^*$ fine structure and this amenable coding significantly simplifies the theory. Towards the end of the first chapter, we prove the full condensation lemma for premice with Mitchell-Steel indexing. This was originally proven by Jensen for premice with $\lambda$-indexing. The second chapter is devoted to the proof of the above mentioned covering theorem.