Take the tour ×  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 t