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?



How elementary can we go?
The hypothesis that $V\_\kappa$ is $\Sigma\_k$ elementary or
even fully elementary in $V$ is much weaker than you say.


One can see part of this quite easily by observing that for
any inaccessible cardinal $\delta$, then
$V\_\delta\models\text{ZFC}$ and there are a club of
ordinals $\alpha$ with $V\_\alpha\prec V\_\delta$. In
particular, if $\delta$ is Mahlo, then there are a
stationary set of inaccessible cardinals $\kappa$ with
$V\_\kappa$ fully elementary in $V\_\delta$.


In particular, if we lived inside $V\_\delta$, we would
believe that there is a stationary proper class of
inaccessible cardinals $\kappa$ with $V\_\kappa$ as fully
elementary in the universe as desired.


It turns out that although we can express
$V\_\kappa\prec\_{\Sigma\_k} V$ as a first-order assertion of
$\kappa$ and $k$, it is not possible to express full
elementary $V\_\kappa\prec V$ as a single first-order
assertion of set theory. Instead, we may use a scheme.


Thus, we introduce $\kappa$ as a constant symbol, and
consider the scheme, denoted "$V\_\kappa\prec V$ ", asserting
of every formula $\varphi$ that $$\forall x\in V\_\kappa\
(\varphi(x) \iff V\_\kappa\models\varphi[x]\ ).$$ If we
add the assumption that $\kappa$ is inaccessible, then this
is known as the Levy scheme.


**Theorem.** The following are equiconsistent over ZFC.


* The Levy scheme. That is, the scheme "$V\_\kappa\prec
 V$ " plus "$\kappa$ is inaccessible."
* "ORD is Mahlo". That is, the scheme asserting of every
definable (with parameters) proper class club, that it
contains an inaccessible cardinal.


Proof. The first implies that $V\_\kappa$ satisfies ORD is
Mahlo, since $\kappa$ will be a limit point and hence an
element of any such club as defined in $V$ using parameters
below $\kappa$. If the second is consistent, then so is the
first by a compactness argument, using the reflection
theorem. QED


Meanwhile, if you drop the inaccessibility requirement,
then you can attain the following, which many set theorists
find surprising.


**Theorem.** The scheme "$V\_\kappa\prec V$ " is
equiconsistent merely with ZFC.


Proof. If ZFC is consistent, then so is every finite
fragment of the scheme $V\_\kappa\prec V$, by the reflection
theorem. QED


One can even attain a proper class club
$C\subset\text{ORD}$ of cardinals, with each $\kappa\in C$
satisfying the scheme $V\_\kappa\prec V$, without going
beyond ZFC in consistency strength.


Both versions of the axiom $V\_\kappa\prec V$ were important
in [my paper on the maximality
principle](http://de.arxiv.org/abs/math.LO/0009240), the
principle asserting that any statement that is forceable in
such a way that it remains true in all further extensions
is already true. It turned out that one can force the
maximality principle only from a model of $V\_\kappa\prec V$
(and you need $\kappa$ inaccessible for the boldface
maximality principle).


