Background construction for $\lambda$-indexed mice

TL;DR

This paper establishes an equivalence between different iteration notions for $\lambda$-indexed mice and introduces a background construction that incorporates Woodin cardinals, advancing the theory of inner models with large cardinals.

Contribution

It proves the equivalence of standard and Mitchell-Steel iteration rules for $\lambda$-indexed mice and develops a background construction that absorbs Woodin cardinals.

Findings

Equivalence of standard and Mitchell-Steel iteration rules for $\lambda$-indexed mice.

Development of a background construction for $\lambda$-indexed mice.

Insights into the correspondence between different iteration trees.

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.