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?
Best Answer
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.
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, 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).