## Notre Dame Journal of Formal Logic

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

Farmer Schlutzenberg

#### Abstract

A significant open problem in inner model theory is the analysis of $\mathrm{HOD}^{L[x]}$ 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 $\delta,\varepsilon$, respectively, such that $M|\delta$ and $N|\varepsilon$ are in $L[x]$, $(\delta^{+})^{M}$ and $(\varepsilon^{+})^{N}$ are countable in $L[x]$, and the pseudo-comparison of $M$ with $N$ succeeds, is in $L[x]$, and lasts exactly $\omega_{1}^{L[x]}$ 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
Accepted: 21 September 2016
First available in Project Euclid: 12 October 2018

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

Digital Object Identifier
doi:10.1215/00294527-2018-0012

Mathematical Reviews number (MathSciNet)
MR3871903

Zentralblatt MATH identifier
06996546

#### 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

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