Answer by Zhen Lin for What axioms need to be added to second-order ZFC before it has a unique model (up to isomorphism)?

My suggestion is essentially the same as Asaf's, but phrased a little differently to avoid directly mentioning inaccessible cardinals etc.

A well-known way of making an axiomatisation of a structure categorical is to add a suitable second-order induction axiom – for instance, second-order arithmetic. So the obvious thing to try in set theory is second-order $\in$-induction:

If $U \subseteq \mathbf{V}$, and $(\forall x \in \mathbf{V} . \, x \in^{\mathbf{V}} y \to x \in U) \to y \in U$, then $U = \mathbf{V}$.

Unfortunately this doesn't actually help here: it just asserts that $\in^\mathbf{V}$ is well-founded, and so if the "real" universe has lots of inaccessibles, then there will be lots of non-isomorphic models satisfying this axiom.

The problem, as far as I can tell, is that this axiom doesn't imply that $\mathbf{V}$ can be constructed inductively from a finite set of operations. In words, it simply says:

If $U$ is a subset of $\mathbf{V}$, such that $\emptyset^\mathbf{V}$ is a member of $U$, and for all elements $y$ of $\mathbf{V}$, if $y$ is a subset of $U$ (in the appropriate sense), then $y$ is a member of $U$ as well.

To put it in type-theoretic terms, the difficulty comes from the fact that the constructors for elements of $\mathbf{V}$ are themselves parametrised by $\mathbf{V}$. So let's think about the fundamental ways of constructing sets:

1. There are two constants: $\emptyset^\mathbf{V}$ and $\omega^\mathbf{V}$.
2. If $x$ and $y$ are elements of $\mathbf{V}$, then so is $\{ x, y \}^{\mathbf{V}}$.
3. If $x$ is an element of $\mathbf{V}$, then so is $\bigcup^{\mathbf{V}} x$.
4. If $x$ is an element of $\mathbf{V}$, then so is $\mathscr{P}^{\mathbf{V}}(x)$.
5. If $x$ is an element of $\mathbf{V}$ and $C$ is a subset of $\mathbf{V}$, then $\{ y : y \in^\mathbf{V} x \land y \in C \}^\mathbf{V}$ (or $\mathsf{sep}^\mathbf{V}(C, x)$ for short) is also an element of $\mathbf{V}$.
6. If $x$ is an element of $\mathbf{V}$, and $F : \mathbf{V} \rightharpoonup \mathbf{V}$ is a partial function, then so is $\{ F (y) : y \in^\mathbf{V} x \land y \in \operatorname{dom} F \}^\mathbf{V}$ (or $\mathsf{repl}^\mathbf{V}(F, x)$ for short).

This suggests the following induction principle:

If $U$ is a subset of $\mathbf{V}$ and $U$ is closed under the above-mentioned constructors, then $U = \mathbf{V}$.

Informally, we are declaring that everything in $\mathbf{V}$ must be constructed using one of the above rules, and this definitely precludes the possibility of (uncountable) inaccessible cardinals in $\mathbf{V}$. Thus, by Mostowski collapse, any model of second-order ZF satisfying the above induction principle must be isomorphic to $(V_\kappa, \in)$, where $\kappa$ is the least inaccessible cardinal.

One also notes that we can economise a little and drop the $\mathsf{sep}^\mathbf{V}$ constructor without changing anything; what remains is more-or-less the definition of Grothendieck universe, at least once we erase all the $\mathbf{V}$ superscripts and add the requirement of transitivity.