Let a cardinal $\kappa$ be $n$-shadow iff $\kappa$ does not have cofinality $\omega$ and for any $n$-th order sentence $\varphi$ in the language $\mathcal{L}_\in$, $\varphi\Leftrightarrow V_\kappa\models\varphi$. If this is unclear, for a standard $M\models\text{ZFC}$ and some $x\in M$ which is a cardinal in $M$, $x$ is $n$-shadow in $M$ iff $V_{\kappa+n}\cap M$ is elementarily equivalent to $\mathcal{P}^n (M)$.

The property of being $0$-shadow is (much) weaker than being Mahlo; in fact, much weaker than $\text{Ord is Mahlo}$.

The property of being $n$-shadow is also (much, much, much) weaker than being $n$-extendible. Furthermore, if there is an $n$-extendible cardinal, then $\text{Con}(\text{ZFC}^2+n\text{-shadow})$.

The reason I bring this up is because of the interesting properties of $1$-shadow cardinals when given an inner model $M$ of ZFC.

Assuming $M$ is an inner model and there is a $1$-shadow cardinal in $M$, then there is no nontrivial elementary embedding from $M$ into itself (in most cases this is equivalent to it's sharp not existing).

So, if $0^{\#}$ exists then there are no $1$-shadow cardinals in $L$. If $0^{\dagger}$ exists then there are no $1$-shadow cardinals in $L[U]$ for the standard $U$.

Specifically, if there is a nontrivial elementary embedding from $K$ into itself, then no cardinal is $1$-shadow in $K$, even though it should be that every uncountable cardinal has most large cardinal properties in $K$ if such an embedding exists, because of the tendencies of the core models.

**Questions: What is the consistency strength of $n$-shadow cardinals? What properties result from these properties of inner models?**