A long pseudo-comparison of premice in $L[x]$

TL;DR

This paper investigates the pseudo-comparison process of premice within $L[x]$, revealing obstacles to analyzing $ ext{HOD}^{L[x]}$ as a core model under large cardinal assumptions, with specific constructions involving Woodin cardinals.

Contribution

It introduces a long pseudo-comparison phenomenon of premice in $L[x]$, demonstrating limitations in core model analysis with large cardinals and specific inner model constructions.

Findings

Pseudo-comparison can last through $oldsymbol{ ext{omega}_1^{L[x]}}$ stages.

Existence of premice with specific properties that cause long pseudo-comparisons.

Obstacles to analyzing $ ext{HOD}^{L[x]}$ as a core model under large cardinal assumptions.

Abstract

We describe an obstacle to the analysis of $\mathrm{HOD}^{L[x]}$ as a core model: Assuming sufficient large cardinals, for a Turing cone of reals $x$ there are premice $M,N$ in $\mathrm{HC}^{L[x]}$ such that the pseudo-comparison of $L[M]$ with $L[N]$ succeeds, is computed in $L[x]$, and lasts through $\omega_1^{L[x]}$ stages. Moreover, we can take $M=M_1|(\delta^+)^{M_1}$ where $M_1$ is the minimal iterable proper class inner model with a Woodin cardinal, and $\delta$ is that Woodin. We can take $N$ such that $L[N]$ is $M_1$-like and short-tree-iterable.