Measures in Mice

TL;DR

This thesis investigates measures in fine structural mice, extending known results to mice below superstrong cardinals and applying the analysis to show that certain tame mice satisfy V=HOD and characterize Suslin sets.

Contribution

It generalizes Kunen's measure uniqueness result to mice below superstrong cardinals and proves that tame mice satisfy V=HOD, providing new insights into their structure.

Findings

Unique measures in inner models for one measurable cardinal

Tame mice satisfy V=HOD

All homogeneously Suslin sets in M_n are Δ^1_{n+1}

Abstract

This thesis analyses extenders in fine structural mice. Kunen showed that in the inner model for one measurable cardinal, there is a unique measure. This result is generalized, in various ways, to mice below a superstrong cardinal. The analysis is then used to show that certain tame mice satisfy $V=\mathsf{HOD}$. In particular, the approach proides a new proof of this result for the inner model $M_n$ for $n$ Woodin cardinals. It is also shown that in $M_n$, all homogeneously Suslin sets of reals are $\mathbf{\Delta}^1_{n+1}$.