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

TL;DR

Under the assumption of _3-determinacy, the paper proves that for certain reals, the HOD of L[x] models satisfies GCH and aligns with a Jensen-Steel core model up to a specific ordinal.

Contribution

It demonstrates that _3-determinacy implies HOD^{L[x]} is a Jensen-Steel core model up to ^{L[x]}, extending understanding of inner models under determinacy assumptions.

Findings

HOD^{L[x]} models GCH under _3-determinacy.

HOD^{L[x]} is a Jensen-Steel core model up to ^{L[x]}.

Results connect determinacy assumptions with inner model theory.

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 $\omega_2^{L[x]}$.