I was reading Kanamori's *The Higher Infinite*, when I came across the fact that for any extendible cardinal $\kappa$ and any $\mathcal{L}_{\kappa,\kappa}^n$-theory $T$, $Sat(T)\Leftrightarrow \forall t\subseteq T(|t|<\kappa\rightarrow Sat(t))$

I thought this was interesting, as any $\mathcal{L}_{\omega,\omega}^n$-theory $T$ (that is, normal $n+1$-th order logic) is also a $\mathcal{L}_{\kappa,\kappa}^n$-theory. This implies that, assuming an extendible cardinal exists, the strong compactness cardinal of $n+1$-th order logic exists and is at most the least extendible cardinal. In fact, the strong compactness cardinal of normal logic, the union of all $n$-th order logics for $0<n<\omega$, is at most the least extendible cardinal.

This brings up the following questions: **What is the consistency strength of asserting that $n$-th order logic has a strong compactness cardinal (for $n>1$)? If there is a strong compactness cardinal of $n$-th order logic, what large cardinal properties does it have?**

*EDIT:* I have since done some thinking on this problem, and realized the strong compactness cardinal of "normal logic" (as I called it) is at least the supremum of all strong compactness cardinals of all $\mathcal{L}_{\kappa,\kappa}^n$ for natural $n$ and cardinals $\kappa$. This of course means that, letting the strong compactness cardinal of normal logic be $\lambda$, the strong compactness cardinal of $\mathcal{L}_{\lambda,\lambda}^n$ is at most $\lambda$. Because of this, $\lambda$ must be extendible.

This fact combined with the fact the the strong compactness cardinal of normal logic is at most the least extendible cardinal shows that **the strong compactness cardinal of normal logic is in fact the least extendible cardinal.**