If $\kappa$ is an inaccessible cardinal then $V\_\kappa$ is a model of $\sf ZFC\_2$ ($\sf ZFC$ with a second-order replacement axiom).


If there are many inaccessible cardinals then there are many models of $\sf ZFC\_2$, but one can add all sort of $\varphi$ which describe $V\_\kappa$ completely. For example if $\varphi$ is "there is no inaccessible cardinal" then $\sf ZFC\_2+\varphi$ is only satisfied by $V\_\kappa$ if $\kappa$ is the least inaccessible cardinal. We can continue and state that there is exactly one, or two, or $\alpha$ inaccessible cardinals for every "small enough" $\alpha$. We can continue by adding more and more information (e.g. there is no inaccessible which is the limit of inaccessibles; there is only one inaccessible which has a stationary set of inaccessibles; and so on).


Let us say that $\varphi$ in $n$-order logic is a categorizer for $\kappa$ if $V\_\kappa$ is the unique model of $\sf ZFC\_2+\varphi$. We say that $\kappa$ is **$\Pi^n\_m$-categorical** (or $\Sigma^n\_m$-categorical) if there is a $\Pi^n\_m$ ($\Sigma^n\_m$) sentence $\varphi$ which is a categorizer for $\kappa$.


On the other hand, we say that $\kappa$ is $\Pi^n\_m$-indescribable if for every $R\subseteq V\_\kappa$ and a $\Pi^n\_m$ sentence $\psi$ such that $\langle V\_\kappa,\in,R\rangle\models\psi$ there is some $\alpha<\kappa$ such that $\langle V\_\alpha,\in,R\cap V\_\alpha\rangle\models\psi$. (Similarly, of course, we define $\Sigma^n\_m$-indescribable cardinals.)


Note that inaccessible cardinals are $\Pi^1\_0$-indescribable, but the least inaccessible is $\Pi^0\_n$-categorical for some $n$ (because the statement "there are no inaccessible cardinals is a first-order statement).



> 
> **Question:** Is there a [deep?] connection between the two notions?
> 
> 
> 



Indescribability of cardinals and categoricity of $V_\kappa$
Categoricity should perhaps be conceived of not as a large
cardinal notion, but rather as an *anti*-large cardinal notion,
since most large cardinal concepts express some degree of
reflection, which is the opposite of categoricity.


For example, if $\kappa$ is $\Pi^n\_m$-categorical, then clearly it
is not $\Pi^n\_m$-indescribable. This is one connection between
your concepts.


But let me argue that there cannot be an equivalence here between
levels of categoricity and failure of indescribability. First, let's observe the following.


**Lemma.** Every $\Pi^n\_m$-categorical cardinal is
$\Delta\_2$-definable in set theory.


Proof. If $\kappa$ is $\Pi^n\_m$-categorical, using formula
$\varphi$, then $\kappa$ is the unique cardinal such that
$V\_\kappa$ satisfies $\varphi$, and this is a first-order
expressible fact $\varphi^\ast$ about $V\_{\kappa+n}$ (since the
second order quantifiers of $\varphi$ are interpreted as intended
as first-order quantifiers inside $V\_{\kappa+n}$. So $\kappa$ is
the unique ordinal such that $V\_{\kappa+n}\models\varphi^\ast$.
This is $\Pi\_2$ expressible as follows: a given ordinal $\xi$ is
the $\kappa$ we are talking about if and only if $\forall Z\ (Z=V\_{\xi+n}\to Z\models\varphi^\ast)$. This is a $\Pi\_2$
assertion, since saying that $Z=V\_{\xi+n}$ has complexity $\Pi\_1$---one must say that $Z$ computes the power sets of its members
correctly---and the latter part has all quantifiers bounded by $Z$.
Similarly, $\xi=\kappa$ if and only if $\exists Z\ Z=V\_{\xi+n}$
and $Z\models\varphi^\ast$, which is a $\Sigma\_2$ formulation. QED


Now, recall that a cardinal $\delta$ is *$\Sigma\_2$-correct*, if
$V\_\delta\prec\_{\Sigma\_2} V$. The reflection theorem shows that
there is a closed unbounded class of $\Sigma\_2$-correct cardinals.
A *$\Sigma\_2$-reflecting* cardinal is a regular $\Sigma\_2$-correct
cardinal, and these have the consistency strength strictly weaker
than a Mahlo cardinal. Meanwhile, every strong cardinal, every
strongly unfoldable cardinal (a transfinite continuation of the
totally indescribable cardinals), every supercompact cardinal is
$\Sigma\_2$-reflecting.


**Theorem.** Every $\Pi^n\_m$-categorical cardinal is smaller than
every $\Sigma\_2$-correct ordinal.


Proof. If $\kappa$ is $\Pi^n\_m$-categorical, then it is
$\Sigma\_2$-definable by some formula $\psi$. In particular, the
assertion "$\exists\kappa\ \psi(\kappa)$" is true in $V$. Since
this is a $\Sigma\_2$-assertion, it follows that if $\delta$ is
$\Sigma\_2$-correct, then $V\_\delta$ will agree that there is such
a $\kappa$. Since $\kappa$ is unique with $\psi(\kappa)$, it
follows that $\kappa\lt\delta$. QED


So from a large cardinal perspective, the categorical cardinals must lie low in the hierarchy of
ordinals.


Notice that since there are only countably many formulas,
and in set theory we have a uniformly expressible account of
whether a cardinal is $\Pi^n\_m$-categorical by a given formula, it follows that there
will be only countably many $\Pi^n\_m$-categorical cardinals. True
reflection does not even begin until you are above them all. But
meanwhile, the indescribability and non-indescribability phenomena
are unbounded in the ordinals.


Lastly, let me mention that the [strongly unfoldable cardinals](http://jdh.hamkins.org/tag/unfoldable-cardinals/) are best understood as a transfinite extension of the indescribability notions of indescribable cardinals. That is, they in effect replace $\Pi^n\_m$ with $\Pi^\alpha\_m$ for transfinite $\alpha$. But at this level (and even at the finite levels in my opinion), it is better to characterize the property in terms of embeddings than higher-order logic. 


