Of Mice and Machetes

TL;DR

This paper explores the construction of certain inner models using iterated mice and Prikry forcings, revealing their relationship to models built from regular cardinals and hyperclass iterations.

Contribution

It demonstrates that models like $L[R]$ can be generated by iterating a small 'machete' mouse and applying hyperclass Magidor iterations, connecting mice with inner models based on regular cardinals.

Findings

Inner models $L[R]$ can be constructed via small mice and Prikry forcings.

Simple mice are elements of $L[Reg]$, linking mice to models built from regular cardinals.

The approach unifies mice construction with inner model theory involving hyperinaccessible cardinals.

Abstract

Let $R$ be the class of regular cardinals which are not hyperinaccessible. We show that $L[R]$, and similar inner models in the $\alpha$-inaccessible hierarchy, can be generated by iterating a small "machete" mouse up through all the ordinals, and then taking a generic extension by a hyperclass Magidor iteration of Prikry forcings. We then show that such simple mice are themselves elements of $L[\mathsf{Reg}]$.