TITLE: How elementary can we go?
QUESTION [13 upvotes]: It is a theorem of A. Levy, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1} V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ sentences.
One might expect that the "amount" of elementarity will grow quickly as we progress with large cardinal axioms, however for the next step, $V_\kappa\prec_{\Sigma_2}V$ we need to get much higher. In order to assure this level of elementarity a supercompact is enough (is it too strong? judging by the stage this theorem appears in Jech's and Kanamori's textbooks I would say that if it is too strong then it is not strong by that much)
To have $\Sigma_3$ we need to go even further to extendible cardinals (again, this might be too strong. I am not too familiar with this notion yet).

Is there a known large cardinal notion to give $\Sigma_4$ elementarity of $V_\kappa$? What about larger $n$? 
I would expect complete elementarity to fail due to some Kunen inconsistency theorem sort of argument, is this true?
Are there results in the reverse direction? Namely if $\kappa$ is such that $V_\kappa\prec_{\Sigma_k}V$ then $\kappa$ has to be inaccessible/supercompact/extendible/etc

If we use all sort of set theoretic notions to measure how far $V$ is from an inner model (forcing axioms, large cardinals, how the cardinals behave in the inner model compared to $V$, sharps and covering theorems, etc etc).
Assuming the answer to the first question is not "It is inconsistent.", is there a useful way to use this approach to measure the difference between $V$ and its inner models?

REPLY [3 votes]: I would like to mention, on the second question, that superhuge cardinals are in fact $\Sigma_5$-reflecting, that is to say inaccessible and $V_\kappa\prec_{\Sigma_5} V$. The reason is first that every superhuge cardinal is extendbile, and so $\Pi_4$ formulas are downward absolute in $V_\kappa$. Suppose we have $\exists x(\phi(x,x_0,\dots,x_n))$, where $\phi(x,x_0,\dots,x_n)$ is $\Pi_4$. Let $z$ be a witness to that, and let $\lambda$ be $\Pi_4$-reflecting such that $z\in V_\lambda$. Then, we can find some $j\colon V\rightarrow M$ such that $M\vDash(\Pi_4\text{ formulas are downward absolute in }V_{j(\kappa)})$ and $j(\kappa)>\lambda$. Then $z\in V_{j(\kappa)}$, and so $M\vDash(V_{j(\kappa)}\vDash\exists x(\phi(x,x_0,\dots,x_n)))$, and so $V_{j(\kappa)}\vDash\exists x(\phi(x,x_0,\dots,x_n))$. As $V_\kappa\prec V_{j(\kappa)}$, we have $V_\kappa\vDash\exists x(\phi(x,x_0,\dots,x_n))$ and so $V_\kappa\prec_{\Sigma_5} V$.
In fact, the set of such cardinals form a normal measure below $\kappa$. Let $D=\{X\subseteq\kappa\mid\kappa\in j(X)\}$. Then by a similar argument $M\vDash(\text{$\kappa$ is $\Sigma_5$-reflecting})$, then $U\in D$,  where $U=\{\lambda<\kappa\mid\text{$\lambda$ is $\Sigma_5$-reflecting}\}$.