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Answering this question would either require refuting choiceless large cardinals or getting close to refuting Woodin's HOD Conjecture.

First, if choiceless cardinals are consistent, one cannot rule out the existence of a second-order elementary embedding. Corollary 4.8 in Usuba's "A note on Lowenheim-Skolem cardinals" states that if there is a proper class of Lowenheim-Skolem cardinals, one can force the Axiom of Choice by a homogeneous definable class forcing. (He doesn't say homogeneous, but it will be.) Therefore assume there is a $j : V\to V$ and there is a proper class of Lowenheim-Skolem cardinals. Let $\mathbb P$ be Usuba's forcing, and let $G\subseteq \mathbb P$ be a generic filter. Then $j\restriction \text{Ord}$ is second-order elementary in $V[G]$: if $\varphi(\vec v)$ is a second-order formula and $V[G]\vDash \varphi(\vec \alpha)$, then by homogeneity, $1_\mathbb P\Vdash \varphi(\vec \alpha)$, and so by the definability of $\mathbb P$ and the elementarity of $j$, $1_\mathbb P\Vdash \varphi(j(\vec\alpha))$, and hence $V[G]\vDash \varphi(j(\vec\alpha))$.

Second, assuming the HOD Conjecture, one can rule out the existence of a nontrivial second-order elementary embedding of the ordinals with critical point above the least extendible cardinal. This follows from Corollary 25 of Woodin-Davis-Rodríguez's paper "The HOD Dichotomy." The reason is that any second-order elementary embedding $j : \text{Ord}\to\text{Ord}$ extends to an elementary embedding from $\text{HOD}$ to $\text{HOD}$. It is also a prominent open question whether such an elementary embedding can exist, even assuming strong versions of the HOD Conjecture.

Answering this question would either require refuting choiceless large cardinals or getting close to refuting Woodin's HOD Conjecture.

First, if choiceless cardinals are consistent, one cannot rule out the existence of a second-order elementary embedding. Corollary 4.8 in Usuba's "A note on Lowenheim-Skolem cardinals" states that if there is a proper class of Lowenheim-Skolem cardinals, one can force the Axiom of Choice by a homogeneous definable class forcing. (He doesn't say homogeneous, but it will be.) Therefore assume there is a $j : V\to V$ and there is a proper class of Lowenheim-Skolem cardinals. Let $\mathbb P$ be Usuba's forcing, and let $G\subseteq \mathbb P$ be a generic filter. Then $j\restriction \text{Ord}$ is second-order elementary in $V[G]$: if $\varphi(\vec v)$ is a second-order formula and $V[G]\vDash \varphi(\vec \alpha)$, then by homogeneity, $1_\mathbb P\Vdash \varphi(\vec \alpha)$, and so by the definability of $\mathbb P$ and the elementarity of $j$, $1_\mathbb P\Vdash \varphi(j(\vec\alpha))$, and hence $V[G]\vDash \varphi(j(\vec\alpha))$.

Second, assuming the HOD Conjecture and the existence of an extendible cardinal, one can rule out the existence of a nontrivial second-order elementary embedding of the ordinals. This follows from Theorem 3.47 of Woodin's paper "In search of Ultimate L." The reason is that any second-order elementary embedding $j : \text{Ord}\to\text{Ord}$ extends to an elementary embedding from the structure $(\text{HOD},T)$ to $(\text{HOD},T)$ where $T$ is the $\Sigma_2$ theory of $V$ with ordinal parameters. It is a prominent open question whether such an elementary embedding can ever exist. It is also open whether there can be an elementary embedding from $\text{HOD}$ to $\text{HOD}$ in the context of the HOD Conjecture.

So if strong choiceless large cardinals are consistent, so is a second-order embedding, while if the HOD Conjecture is true, second-order embeddings are ruled out by more compelling large cardinal axioms. The question is therefore just one aspect of one of the most important questions in set theory/large cardinals: to what extent can arbitrary sets be approximated by ordinal definable ones?

Here's a rough answer that I'll clean up after seminar. Corollary 4.8 in Usuba's "A note on Lowenheim-Skolem cardinals" states that if there is a proper class of Lowenheim-Skolem cardinals, one can force the Axiom of Choice by a homogeneous definable class forcing. (He doesn't say homogeneous, but it will be.) Therefore assume there is a $j : V\to V$ and there is a proper class of Lowenheim-Skolem cardinals. Let $\mathbb P$ be Usuba's forcing, and let $G\subseteq \mathbb P$ be a generic filter. Then $j\restriction \text{Ord}$ is second-order elementary in $V[G]$: if $\varphi(\vec v)$ is a second-order formula and $V[G]\vDash \varphi(\vec \alpha)$, then by homogeneity, $1_\mathbb P\Vdash \varphi(\vec \alpha)$, and so by the definability of $\mathbb P$ and the elementarity of $j$, $1_\mathbb P\Vdash \varphi(j(\vec\alpha))$, and hence $V[G]\vDash \varphi(j(\vec\alpha))$.

It is reasonable to doubt the consistency of the theory NBG + there is a Reinhardt cardinal + there is a proper class of Lowenheim-Skolem cardinals. Assuming the HOD Conjecture, one can rule out the existence of a nontrivial second-order elementary embedding of the ordinals with critical point above the least extendible cardinal. This follows from Corollary 25 of Woodin-Davis-Rodríguez's paper "The HOD Dichotomy."

Answering this question would either require refuting choiceless large cardinals or getting close to refuting Woodin's HOD Conjecture.

First, if choiceless cardinals are consistent, one cannot rule out the existence of a second-order elementary embedding. Corollary 4.8 in Usuba's "A note on Lowenheim-Skolem cardinals" states that if there is a proper class of Lowenheim-Skolem cardinals, one can force the Axiom of Choice by a homogeneous definable class forcing. (He doesn't say homogeneous, but it will be.) Therefore assume there is a $j : V\to V$ and there is a proper class of Lowenheim-Skolem cardinals. Let $\mathbb P$ be Usuba's forcing, and let $G\subseteq \mathbb P$ be a generic filter. Then $j\restriction \text{Ord}$ is second-order elementary in $V[G]$: if $\varphi(\vec v)$ is a second-order formula and $V[G]\vDash \varphi(\vec \alpha)$, then by homogeneity, $1_\mathbb P\Vdash \varphi(\vec \alpha)$, and so by the definability of $\mathbb P$ and the elementarity of $j$, $1_\mathbb P\Vdash \varphi(j(\vec\alpha))$, and hence $V[G]\vDash \varphi(j(\vec\alpha))$.

Second, assuming the HOD Conjecture, one can rule out the existence of a nontrivial second-order elementary embedding of the ordinals with critical point above the least extendible cardinal. This follows from Corollary 25 of Woodin-Davis-Rodríguez's paper "The HOD Dichotomy." The reason is that any second-order elementary embedding $j : \text{Ord}\to\text{Ord}$ extends to an elementary embedding from $\text{HOD}$ to $\text{HOD}$. It is also a prominent open question whether such an elementary embedding can exist, even assuming strong versions of the HOD Conjecture.

Here's a rough answer that I'll clean up after seminar. A theorem of Usuba states that if there is a proper class of Lowenheim-Skolem cardinals, one can force the Axiom of Choice by a homogeneous definable class forcing. Therefore assume there is a $j : V\to V$ and there is a proper class of Lowenheim-Skolem cardinals. Let $\mathbb P$ be Usuba's forcing, and let $G\subseteq \mathbb P$ be a generic filter. Then $j\restriction \text{Ord}$ is second-order elementary in $V[G]$: if $\varphi(\vec v)$ is a second-order formula and $V[G]\vDash \varphi(\vec \alpha)$, then by homogeneity, $1_\mathbb P\Vdash \varphi(\vec \alpha)$, and so by the definability of $\mathbb P$ and the elementarity of $j$, $1_\mathbb P\Vdash \varphi(j(\vec\alpha))$, and hence $V[G]\vDash \varphi(j(\vec\alpha))$.

It is reasonable to doubt the consistency of the theory NBG + there is a Reinhardt cardinal + there is a proper class of Lowenheim-Skolem cardinals. Assuming the HOD Conjecture, one can rule out the existence of such a nontrivial second-order elementary embedding with critical point above the least extendible cardinal. This follows from Corollary 25 of Woodin-Davis-Rodríguez's paper "The HOD Dichotomy."

Here's a rough answer that I'll clean up after seminar. Corollary 4.8 in Usuba's "A note on Lowenheim-Skolem cardinals" states that if there is a proper class of Lowenheim-Skolem cardinals, one can force the Axiom of Choice by a homogeneous definable class forcing. (He doesn't say homogeneous, but it will be.) Therefore assume there is a $j : V\to V$ and there is a proper class of Lowenheim-Skolem cardinals. Let $\mathbb P$ be Usuba's forcing, and let $G\subseteq \mathbb P$ be a generic filter. Then $j\restriction \text{Ord}$ is second-order elementary in $V[G]$: if $\varphi(\vec v)$ is a second-order formula and $V[G]\vDash \varphi(\vec \alpha)$, then by homogeneity, $1_\mathbb P\Vdash \varphi(\vec \alpha)$, and so by the definability of $\mathbb P$ and the elementarity of $j$, $1_\mathbb P\Vdash \varphi(j(\vec\alpha))$, and hence $V[G]\vDash \varphi(j(\vec\alpha))$.

It is reasonable to doubt the consistency of the theory NBG + there is a Reinhardt cardinal + there is a proper class of Lowenheim-Skolem cardinals. Assuming the HOD Conjecture, one can rule out the existence of a nontrivial second-order elementary embedding of the ordinals with critical point above the least extendible cardinal. This follows from Corollary 25 of Woodin-Davis-Rodríguez's paper "The HOD Dichotomy."