${\Sigma }_1$ gaps as derived models and correctness of mice

Abstract

Assume ZF + AD + V=L(R). Let $[\alpha ,\beta ]$ be a ${\Sigma }_1$ gap with $J_{\alpha }(R)$ admissible. We analyze $J_{\beta }(R)$ as a natural form of "derived model" of a premouse $P$, where $P$ is found in a generic extension of $V$. In particular, we will have $\mathcal{P}(R)\cap J_{\beta }(R)=\mathcal{P}(R)\cap D$, and if $J_{\beta }(R)\models$ "$\Theta$ exists", then $J_{\beta }(R)$ and $D$ in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of $(L(R){)}^M$, for $\omega$-small mice $M$. We also establish some preliminary work toward this conjecture in the present paper.

128 pages. Added acknowledgements, updated references