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

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.