[Paper Review] Measures in Mice

Farmer Schlutzenberg|arXiv (Cornell University)|Jan 20, 2013

Advanced Topology and Set Theory17 references18 citations

TL;DR

This paper generalizes Kunen's result on unique measures in inner models to mice below a superstrong cardinal, using structural analysis of extenders to prove that certain tame mice satisfy $V = \mathsf{HOD}$. It provides a new proof for $M_n$, the inner model with $n$ Woodin cardinals, and establishes that all homogeneously Suslin sets of reals in $M_n$ are $\mathbf{\Delta}^1_{n+1}$.

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 $\mathbf{\Delta}^1_{n+1}$.

Motivation & Objective

• To extend Kunen's result on unique measures in the inner model for one measurable cardinal to mice below a superstrong cardinal.
• To analyze the structure of extenders in fine structural mice to establish properties of $V = \mathsf{HOD}$ in tame mice.
• To provide a new proof that $V = \mathsf{HOD}$ holds in the inner model $M_n$ for $n$ Woodin cardinals.
• To determine the complexity of homogeneously Suslin sets of reals in $M_n$, specifically their classification as $\mathbf{\Delta}^1_{n+1}$.

Proposed method

• Analyzes extenders in fine structural mice below a superstrong cardinal using inner model-theoretic techniques.
• Applies structural properties of mice to generalize Kunen's uniqueness result on measures to broader classes of models.
• Employs fine structure theory to examine the definability and closure properties of sets in $M_n$.
• Uses homogeneously Suslin sets to analyze determinacy and complexity in $M_n$, linking them to $\mathbf{\Delta}^1_{n+1}$-definability.
• Establishes $V = \mathsf{HOD}$ in tame mice by analyzing the interplay between extenders and definable subsets of the universe.
• Applies known results on $M_n$ to derive complexity bounds on sets of reals via measure-theoretic and structural arguments.

Experimental results

Research questions

• RQ1 Can Kunen's uniqueness result on measures in the inner model for one measurable cardinal be generalized to mice below a superstrong cardinal?
• RQ2 Under what conditions do tame mice satisfy $V = \mathsf{HOD}$, and what structural features enable this?
• RQ3 What is the exact complexity of homogeneously Suslin sets of reals in the inner model $M_n$ with $n$ Woodin cardinals?
• RQ4 How does the analysis of extenders in fine structural mice contribute to proving $V = \mathsf{HOD}$ in $M_n$?
• RQ5 What is the relationship between the structure of extenders and the definability of sets of reals in $M_n$?

Key findings

• The paper establishes that in certain tame mice, $V = \mathsf{HOD}$ holds, generalizing known results in inner model theory.
• A new proof is provided for $V = \mathsf{HOD}$ in the inner model $M_n$ for $n$ Woodin cardinals, based on extender analysis.
• All homogeneously Suslin sets of reals in $M_n$ are shown to be $\mathbf{\Delta}^1_{n+1}$, refining their definitional complexity.
• The analysis confirms that extenders in fine structural mice below a superstrong cardinal support unique measures under specific conditions.
• The structural properties of mice enable a uniform treatment of definability and measure uniqueness, extending Kunen's original framework.
• The results demonstrate that the interplay between extender structure and definability leads to strong global consequences like $V = \mathsf{HOD}$.