Indescribability of cardinals and categoricity of $V_\kappa$

Question

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?

Answer 1

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 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.