# Try again. Fail again. Fail better.

I am still unable to fix the mistake in the paper in the previous post.

The method in that paper ends up with a “limit model of $\mathrm{HOD}^{L[x]}$” whose Woodin cardinal is $u_{\omega}$. Therefore, this “limit model” up to its Woodin is “obviously” $L_{u_\omega}[\mathcal{O}_{\Sigma^1_3}]$ (cf. Iterates of $M_1$), but I don’t see why. Maybe it’s just as hard as computing $\mathrm{HOD}^{L[x]}$.

# New paper: The internal structure of $\mathrm{HOD}^{L[x]}$ up to its Woodin

I have posted a new paper on arXiv.

Title: The internal structure of $\mathrm{HOD}^{L[x]}$ up to its Woodin

Abstract: Assume $\boldsymbol{\Delta}^1_3$-determinacy. It is shown that for any $x \geq_T M_1^{\#}$, $\mathrm{HOD}^{L[x]}$ is a model of GCH, and in fact, it is a Jensen-Steel core model up to $\omega_2^{L[x]}$.

# test with formula in title $x^3 + y^3 = z^3$

write a post

An inline formula is $(M_1^{\#})^{\mathcal{N}}$.

A block formula is

A blockquote

Definition. A cardinal $\delta$ is a Woodin cardinal iff for all $A \subseteq V_{\delta}$ there is $\kappa < \delta$ such that $\kappa$ is $< \delta$-$A$-strong.

An old 2048 game extended
(Disclaimer: I am not responsible for your wasted time on this game!)