Background construction for $\lambda$-indexed mice

Abstract

Let $M$ be a $\lambda$-indexed (that is, Jensen indexed) premouse. We prove that $M$ is iterable with respect to standard $\lambda$-iteration rules iff $M$ is iterable with respect to a natural version of Mitchell-Steel iteration rules. Using this equivalence, we describe a background construction for $\lambda$-indexed mice, analogous to traditional background constructions for Mitchell-Steel indexed mice, and which absorbs Woodin cardinals from the background universe. We also prove some facts regarding the correspondence between standard iteration trees and u-iteration trees on premice with Mitchell-Steel indexing.