How expressive can $\mathcal{L}_{\kappa,\kappa}$ be?

Question

(Note that the logic systems described in this question only refer to logic systems restricted to the language $\in$)

1. Can $\mathcal{L}_{\kappa,\kappa}$ express $n$-th order finitary logic? It is clear that it can express first-order logic ($\Pi_{<\omega}^0$), but it seems unlikely that $\mathcal{L}_{\kappa,\kappa}$ can express even $\Pi_{<\omega}^1$.

By the answer to this question: Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem), it is known that $L^{\Pi_{<\omega}^1}=HOD$. Assuming $\mathcal{L}_{\kappa,\kappa}$ expresses $\Pi_{<\omega}^1$, it is then not difficult to show that $HOD\subseteq L^{\mathcal{L}_{\kappa,\kappa}}\subseteq L(V_\kappa)$. Thus, unless $L(V_\kappa)=V$ (which is true iff $L=V$), $V\neq HOD$.

Therefore, the existence of such a $\kappa$ is inconsistent with $V\neq L\land V=HOD$. This is very doubtful.

2. Can $\mathcal{L}_{\kappa,\kappa}$ express $\Pi_n^1$ for $n>0$? For $n=0$, it is once again easy to see that, yes, this is true ($\Pi_0^1=\Pi_0^0$) for every $\kappa$. Other than this, I have no information for this question.

Answer 1

The answer is no.

Suppose that $\kappa$ is a Mahlo cardinal. This is $\Pi^1_1$ expressible in $V_\kappa$, since it amounts to the assertion that every closed unbounded subset of $\kappa$ contains a regular cardinal. But it cannot be expressible by an assertion of $\mathcal{L}_{\kappa,\kappa}$, since there is a forcing extension $V[G]$ in which $V_\kappa$ is preserved, but which destroys the Mahloness of $\kappa$. Since $V_\kappa$ is preserved, the truth of any $\mathcal{L}_{\kappa,\kappa}$ assertion is invariant by the forcing, but the Mahloness of $\kappa$ is not.

One can make other similar kinds of examples. For example, "$\kappa$ is regular" is expressible in $V_\kappa$ by a $\Pi^1_1$ assertion, but it cannot be provably equivalent to any $\mathcal{L}_{\kappa,\kappa}$ assertion, since for example with a measurable cardinal, this can be changed by forcing without changing $V_\kappa$.

Here is another way to argue, without forcing. Let $\kappa$ be an inaccessible cardinal. The assertion that $\kappa$ is regular is $\Pi^1_1$ expressible in $V_\kappa$. If we fix a $\mathcal{L}_{\kappa,\kappa}$ truth predicate for $V_\kappa$, we can find many $\gamma<\kappa$ with $V_\gamma\prec_{\mathcal{L}_{\gamma,\gamma}}V_\kappa$. In particular, there will be such $\gamma$ of cofinality $\omega$ above the size of any particular $\varphi$ in $\mathcal{L}_{\kappa,\kappa}$. So $V_\gamma$ and $V_\kappa$ will agree on the truth of $\varphi$, but not on the assertion that the ordinals are regular.

I believe that one can likely press this further to make an example in ZFC, without any large cardinals.