## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 59, Number 4 (2018), 599-604.

### A Long Pseudo-Comparison of Premice in $L\left[x\right]$

#### Abstract

A significant open problem in inner model theory is the analysis of ${\mathrm{HOD}}^{L\left[x\right]}$ as a strategy premouse, for a Turing cone of reals $x$. We describe here an obstacle to such an analysis. Assuming sufficient large cardinals, for a Turing cone of reals $x$ there are proper class $1$-small premice $M,N$, with Woodin cardinals $\delta ,\epsilon $, respectively, such that $M|\delta $ and $N|\epsilon $ are in $L\left[x\right]$, $({\delta}^{+}{)}^{M}$ and $({\epsilon}^{+}{)}^{N}$ are countable in $L\left[x\right]$, and the pseudo-comparison of $M$ with $N$ succeeds, is in $L\left[x\right]$, and lasts exactly ${\omega}_{1}^{L\left[x\right]}$ stages. Moreover, we can take $M={M}_{1}$, the minimal iterable proper class inner model with a Woodin cardinal, and take $N$ to be ${M}_{1}$-like and short-tree-iterable.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 59, Number 4 (2018), 599-604.

**Dates**

Received: 22 October 2015

Accepted: 21 September 2016

First available in Project Euclid: 12 October 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1539309632

**Digital Object Identifier**

doi:10.1215/00294527-2018-0012

**Mathematical Reviews number (MathSciNet)**

MR3871903

**Zentralblatt MATH identifier**

06996546

**Subjects**

Primary: 03E45: Inner models, including constructibility, ordinal definability, and core models

**Keywords**

inner model ordinal definable comparison

#### Citation

Schlutzenberg, Farmer. A Long Pseudo-Comparison of Premice in $L[x]$. Notre Dame J. Formal Logic 59 (2018), no. 4, 599--604. doi:10.1215/00294527-2018-0012. https://projecteuclid.org/euclid.ndjfl/1539309632