In what follows, $\mathsf{ZCKP}$ refers to the subset of $\mathsf{ZFC}$ consisting of the axioms of Zermelo set theory with choice and foundation ($\mathsf{ZC}$) plus those of Kripke-Platek set theory ($\mathsf{KP}$): equivalently, as these two theories have much overlap, $\mathsf{ZCKP}$ consists of $\mathsf{ZC}$ plus the axiom of replacement for $\Sigma_1$-formulæ (or equivalently, $\mathsf{KP}$ plus choice, foundation, powerset, infinity, and full separation). A "**largish**" cardinal property means, informally, one whose existence is provable in $\mathsf{ZFC}$ but not in $\mathsf{ZCKP}$.

Here is the simplest example of a largish cardinal notion. A theorem of Azriel Lévy (see, e.g., Barwise, *Admissible Sets and Structures* (1975), theorem II.3.5 on page 53 and theorem II.9.1 on page 76) states that for every uncountable cardinal $\kappa$, if $H(\kappa)$ is the set of sets hereditarily of cardinality $<\kappa$, then $H(\kappa)$ is a $1$-elementary submodel of the universe (meaning that every $\Sigma_1$ formula with parameters in $H(\kappa)$ is true [in $V$] iff it is true in $H(\kappa)$). This implies that $H(\kappa)$ satisfies $\Delta_0$-collection (eqvt. $\Sigma_1$-replacement), and if $\kappa$ is a **strong limit**, then $H(\kappa) \models \mathsf{ZCKP}$ (and the converse is clear). In particular, $\mathsf{ZCKP}$ does not prove the existence of strong limit cardinals.

Now I am interested in strengthenings of this condition on $\kappa$ such that the existence of these cardinals is still provable in $\mathsf{ZFC}$. Two obvious candidates are:

$H(\kappa)$ satisfies $\mathsf{ZC}$ plus replacement for $\Sigma_n$ formulæ,

$H(\kappa)$ is an $n$-elementary submodel of the universe (i.e., every $\Sigma_n$ formula with parameters in $H(\kappa)$ is true iff it is true in $H(\kappa)$).

These should at least imply that $\kappa$ is a fixed point of the beth function, so that in fact $H(\kappa) = V_\kappa$. (Perhaps this should be added as a precondition to be worthy of the term "largish cardinal".)

**Edit** (on 2015-12-11, following the answer by Joel David Hamkins): I didn't realize how very different the two notions above are: the first (call them "$\Sigma_n$-replacing" cardinals) is "local" in that it involves only sets from $V_\kappa$ and can thus be expressed as a $\Delta_1$ property of $V_\kappa$, whereas the latter ("$\Sigma_n$-correct cardinals") is "global" and involves the entire universe. This has a consequence of size: the smallest $\Sigma_2$-correct cardinal, as explained in Joel's response, is larger than the first $\Sigma_n$-replacing cardinal for all $n$, or even the first inaccessible, etc., and there is no hope of "computing" it. There may be some hope for the first $\Sigma_2$-replacing cardinal, however. I really should have asked two different questions.

*My question is this:* Can these conditions, at least for $n=2$, or perhaps some related ones, be rephrased in purely cardinal-theoretic terms (without appealing to model theory and if possible avoiding the Lévy hierarchy)? Even better, can the smallest cardinal satisfying such a condition be "described" or "computed" in some way? (In the same way that $\beth_\omega$, or "the limit of the sequence defined by $\kappa_0 = \omega$ and $\kappa_{n+1} = \beth_{\kappa_n}$" are descriptions/computations of the smallest strong limit cardinal and the smallest fixed point of the beth function.)

More generally, any comments on these or related properties would be welcome (including a better term than "largish cardinal"). There is probably some connection with powerset-admissible ordinals, although the exact relation escapes me.

One reason why one might be interested in such cardinals is that the corresponding $H(\kappa)$ might serve as a drop-in replacement for Grothendieck universes in a $\mathsf{ZFC}$ formulation of category theory (they are not fully Grothendieck universes, but the point is that the use of a construction that escapes from such a "universish" set is likely to be so rare as to be very conspicuous; and unlike Grothendieck universes, their existence follows from $\mathsf{ZFC}$).