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?