# Measures in Mice

@article{Schlutzenberg2013MeasuresIM, title={Measures in Mice}, author={Farmer Schlutzenberg}, journal={arXiv: Logic}, year={2013} }

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…

## 27 Citations

### Negative results on precipitous ideals on omega_1

- Mathematics
- 2021

We show that in many extender models, e.g. the minimal one with inﬁnitely many Woodin cardinals or the minimal with a Woodin cardinal that is a limit of Woodin cardinals, there are no generic…

### Homogeneously Suslin sets in tame mice

- MathematicsThe Journal of Symbolic Logic
- 2012

Homogeneously Suslin (hom) sets of reals in tame mice are studied and every hom set is correctly , and (δ + 1)-universally Baire where δ is the least Woodin.

### A characterization of extenders of HOD

- Mathematics
- 2019

Assume AD+V = L(R). Let κ = δ ∼ 2 1, the supremum of all ∆∼ 2 1 prewellorderings. We prove that extenders on the sequence of HOD that have critical point κ are generated by countably complete…

### Comparison of fine structural mice via coarse iteration

- MathematicsArch. Math. Log.
- 2014

This work describes a process comparing M with D, and proves that this process succeeds, through forming iteration trees on M and on S, the fine structural mouse.

### Tall Cardinals in Extender Models

- MathematicsNotre Dame J. Formal Log.
- 2021

We obtain a characterization of $\lambda$-tall cardinals in terms of the function $o(\alpha)$ in extender models $L[E]$ which have no inner model with a Woodin caridnal and $L[E] \models \text{``I am…

### The definability of $\mathbb{E}$ in self-iterable mice

- Mathematics
- 2014

Let $M$ be a fine structural mouse and let $F\in M$ be such that $M\models$``$F$ is a total extender'' and $(M||\mathrm{lh}(F),F)$ is a premouse. We show that it follows that $F\in\mathbb{E}^M$,…

### The Ultrapower Axiom

- MathematicsTrends in Set Theory
- 2020

The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The…

### Fine structure from normal iterability.

- Mathematics
- 2020

We show that (i) the standard fine structural properties for premice follow from normal iterability (whereas the classical proof relies on iterability for stacks of normal trees), and (ii) every…

### Hjorth's reflection argument

- Mathematics
- 2021

In [6], Hjorth, assuming AD+ ZF+ DC, showed that there is no sequence of length ω2 consisting of distinct Σ 1 2-sets. We show that the same theory implies that for n ≥ 0, there is no sequence of…

## References

SHOWING 1-10 OF 26 REFERENCES

### Homogeneously Suslin sets in tame mice

- MathematicsThe Journal of Symbolic Logic
- 2012

Homogeneously Suslin (hom) sets of reals in tame mice are studied and every hom set is correctly , and (δ + 1)-universally Baire where δ is the least Woodin.

### Fine structure for tame inner models

- Mathematics, PhilosophyJournal of Symbolic Logic
- 1996

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2]…

### Deconstructing inner model theory

- MathematicsJournal of Symbolic Logic
- 2002

It follows that the coherent sequences constructed in [2] do not satisfy the ISC of [2], and the weaker ISC which these sequences do satisfy is described, and the small changes in the arguments of [1] this new condition requires are indicated.

### Combinatorial principles in the core model for one Woodin

- Mathematics
- 1995

We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form L[B]. We generalize to L[E] some combinatorial principles…

### The Covering Lemma up to a Woodin Cardinal

- MathematicsAnn. Pure Appl. Log.
- 1997

### THE MAXIMALITY OF THE CORE MODEL

- Mathematics
- 1997

Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2) K computes successors of weakly compact cardinals correctly, 3)…

### A minimal counterexample to universal baireness

- MathematicsJournal of Symbolic Logic
- 1999

The construction of a set of reals of minimal complexity is constructed which fails to be universally Baire using a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques.

### An Outline of Inner Model Theory

- Mathematics
- 2010

This paper outlines the basic theory of canonical inner models satisfying large cardinal hypotheses. It begins with the definition of the models, and their fine structural analysis modulo iterability…