Assume that $V\neq HOD$ and let $\kappa = \min \{\alpha\in On \mid \mathcal{P}(\alpha) \not\subseteq HOD\}$. 


Clearly, $\kappa$ is a cardinal. 


**Question:** Is it consistent that $\kappa = \aleph\_\omega$? 


Note that it is consistent that $\kappa$ is a regular cardinal: start with $V=L$ and force with $Add(\kappa,1)$. Since this forcing is weakly homogeneous, its generic filter is not in $HOD$. Since we don't add any bounded subsets to $\kappa$, for every $\alpha < \kappa$, $\mathcal{P}(\alpha) \subseteq L \subseteq HOD$.


Similarly, it is consistent that $\kappa$ is singular with countable cofinality. Let $\kappa$ be a measurable cardinal and let $V = L[\mu]$ ($\mu$ is a normal measure for $\kappa$), the canonical inner model for one measurable cardinal. Let $C$ be a Prikry sequence. Then $HOD^{V[C]}\cap \kappa^{<\kappa} = L[\mu]\cap \kappa^{<\kappa}\subseteq HOD$, but since the Prikry forcing is weakly homogeneous, $C\notin HOD^{V[C]}$.



Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?
Assume $GCH$ and let $\kappa$ be $(\kappa+2)-$strong. Force with extender based Prikry forcing $P$ with interleaved collapses to make $\kappa=\aleph\_\omega$
and $2^{\aleph\_\omega}=\aleph\_{\omega+2}.$ Call the resulting extension $V[H].$
Also let $V[G]$ be an intermediate submodel, which just adds the Prikry sequence related to the normal measure and adds collapses, so that the following holds:


(1) $V[G] \subset V[H]$ have the same cardinals and bounded subsets of $\kappa=\aleph\_\omega,$


(2) $V[G] \models GCH.$


Note that there are many new $\omega$-sequences through $\kappa=\aleph\_\omega$ in $V[H]\setminus V[G].$


A much stronger version of the following lemma will appear in my paper "$HOD, V$ and the $GCH$" (where extender based Prikry forcing is replaced by extender based Radin forcing):


**Homogeneity lemma.** Assume $p,q\in P$ are such that $\pi(p)$ is compatible with $\pi(q),$ where $\pi$ is the projection map. Then there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: P/p' \simeq P/q'.$


It follows from the above lemma that $HOD^{V[H]} \subseteq V[G].$


Now force over $V[H]$ to code any bounded subset of $\aleph\_\omega$ into $HOD$ using a homogeneous forcing, call the extension $V[H][K].$


Now $V[G] \subset V[H][K],$ and any new $\omega-$sequence cofinal in $\kappa$ which is in $V[H]\setminus V[G]$ witnesses $\min\{\alpha: P(\alpha) \nsubseteq HOD\}=\aleph\_\omega$.


**Remark.** In fact it seems we need much weaker assumption. All we need is to be able to add a new cofinal $\omega$-sequence in $\kappa$ which is not in $V[G],$ and it seems to me that for this just a measurable is sufficient.


