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 $\text{Con}(\text{ZFC}^2+n\text{-shadow})$.




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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.




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



A Weak form of Extendibility and Inner Model Theory
You consider the strength of having a $1$-shadow cardinal in an
inner model. But I claim that no proper inner model $M$ has any
$1$-shadow cardinals at all.


Assume $M\subsetneq V$ is a proper inner model, by which I mean
that $M$ is a transitive class model of ZF, containing all
ordinals, and $M\neq V$. Suppose that $\kappa$ is a $1$-shadow
cardinal, which according to your definition means that
$V\_{\kappa+1}\cap M\prec P(M)$. Since $M\neq V$, there is a set
$A\in M$ for which there is a subset $B\subset A$ with $B\in P(M)$
but $B\notin M$. This property is expressible in $P(M)$, whether we
use the full second-order semantics or the GBC semantics, where the
classes of $V$ are given as a collection. But notice that
$V\_{\kappa+1}\cap M$ has no such sets $A$ and $B$, since every
element of $V\_\kappa\cap M$ has all its subsets that are in
$V\_{\kappa+1}\cap M$, since they are all in $M$. $\Box$


This seems to refute your claim that $n$-shadow is necessarily weaker than $n$-extendible, since an inner model can have extendible cardinals. Perhaps the resolution of this issue will be that the definition of the shadow cardinals is not yet expressing exactly what you want to express?


