In what follows, $\mathsf{ZCKP}$ refers to the subset of $\mathsf{ZFC}$ consisting of the axioms of [Zermelo set theory](https://en.wikipedia.org/wiki/Zermelo_set_theory) with choice and foundation ($\mathsf{ZC}$) plus those of [Kripke-Platek set theory](https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_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 $\mathsf{ZFC}$).



"Largish" cardinals
Your cardinals are known as the [$\Sigma\_n$-correct cardinals](http://cantorsattic.info/Reflecting#Reflection_and_correctness), and they arise in diverse set-theoretic contexts. For example, we use them extensively in our paper:


* J. Bagaria, J. D. Hamkins, K. Tsaprounis, T. Usuba, [Superstrong
and other large cardinals are never Laver indestructible](http://jdh.hamkins.org/superstrong-never-indestructible/), to
appear in the Archive for Math Logic.


It is a ZFC theorem that the $\Sigma\_n$-correct cardinals form a
closed unbounded proper class often denoted $C^{(n)}$.


One subtle point about the $\Sigma\_n$-correct cardinals is that
although we have a concept of $\Sigma\_2$-correct and
$\Sigma\_3$-correct and so on, $\Sigma\_n$-correct for any
particular $n$, there is no uniform-in-$n$ way to express the
concept of $\Sigma\_n$-correctness in first-order set theory. The
concept is uniformly expressible in some second-order set
theories, such as Kelley-Morse set theory, which prove that there
is a truth-predicate for first-order truth.


Concerning your question, when $n\geq 2$ there can be no way to
define what it means for $\kappa$ to be $\Sigma\_n$-correct by
looking only below $\kappa$, say, as a limit process, since such a
property would be too simple, as it could be verified inside
$V\_\kappa$ itself, but such verifiable-in-$V\_\kappa$ properties have complexity at worst $\Delta\_2$. For example, the property of being $\Sigma\_n$-correct cannot be
$\Sigma\_n$-expressible, for then the assertion "There is a
$\Sigma\_n$-correct cardinal" would reflect from $V$ to $V\_\kappa$,
even when $\kappa$ is the least $\Sigma\_n$-correct cardinal, which
gives a contradiction since there are none below the least one. Meanwhile, the property of being $\Sigma\_n$-correct is $\Pi\_n$-expressible, since one need only say that all the instances of $\Pi\_n$ truth in $V\_\kappa$ are actually true.


The case of $\Sigma\_2$-correct cardinals is particularly attractive, and perhaps this is an example that interests you. The $\Delta\_2$ properties are precisely the properties that are *local*, in the sense that they can be determined in any sufficiently large $H(\theta)$. You can read more on my blog post: 


* [Local properties in set theory](http://jdh.hamkins.org/local-properties-in-set-theory/)


It follows that a cardinal $\kappa$ is $\Sigma\_2$-correct, if whenever there is an object having a certain properties inside some possibly very large $H(\theta)$, then there is such an object inside such an $H(\theta)$ with $\theta<\kappa$. In other words, $\kappa$ is $\Sigma\_2$-correct, if whenever anything verifiable happens anywhere, then it happens inside $V\_\kappa$. Alternatively, everything verifiable has already happened by the time you get to $H\_\kappa$. Such a way of understanding $\Sigma\_2$-correctness is extremely useful, since it aligns with how set theorists often think about verifying set-theoretic facts. 


(A small matter: the distinction between $V\_\kappa$ and $H\_\kappa$ disappears once $n\geq 2$, since in this case the cardinals are $\beth$-fixed points and so $V\_\kappa=H\_\kappa$.)


Lastly, your idea of using the $\Sigma\_n$-correct cardinals as a
universe replacement idea is well known. This is known as the
[Feferman theory](http://cantorsattic.info/Reflecting#The_Feferman_theory), and I also discussed it here on
MathOverflow in my answer to the question [What interesting/nontrivial results in Algebraic geometry require the existence of universes?](https://mathoverflow.net/a/28913/1946).


