Working in [$\mathcal L\_{\omega\_1, \omega\_1}$](https://plato.stanford.edu/entries/logic-infinitary/), add symbol $=$ with its axioms; add symbol $\in$ and axiomatize:


$\textbf{Extensionality: } \forall x \forall y : \forall z (z \in x \leftrightarrow z \in y) \to x=y$


$\textbf{Foundation: } (\forall v\_n)\_{n \in \omega} \, \exists x: \bigvee\_{n \in \omega} (x=v\_n) \land \bigwedge\_{n \in \omega} (v\_n \not \in x)$


$\textbf{Define: } x=\{y \mid \phi\} \equiv\_{def} \forall y \, (y \in x \leftrightarrow \phi)$


$\textbf{Construction: } \\(\forall v\_n)\_{n \in \omega} \, \exists x : x=\{y \mid \bigvee\_{n \in \omega} ( y=v\_n)\}$


$ \textbf{Countability: } \\ \forall x \, (\exists v\_n)\_{n \in \omega} : x=\{y \mid y \neq v\_0 \land \bigvee\_{n \in \omega} ( y=v\_n)\} $


$\textbf {Empty set: } \exists x \forall y: y \not \in x$.



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> Is this theory Complete?
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> Is this theory Categorical?
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Is this theory of well founded countable sets formalized in infinitary logic, complete and categorical?
Yes, this is categorical and hence complete (with respect to any satisfactory notion of proof, that is). Specifically, I claim that any model $M$ of your theory is isomorphic to $\mathsf{HC}$, the set of hereditarily countable sets.


First, given $M$ we can construct recursively an embedding $i:\mathsf{HC}\rightarrow M$. (Note that this uses the "construction" axiom.) So we just need to show that $i$ is surjective. I think the cleanest way to do this is to note that the image of $i$ is $\mathcal{L}\_{\omega\_1,\omega\_1}$-definable in $M$, as "$a\in im(i)$ iff there is a well-founded countably-branching tree such that, when we recursively label the nodes of $T$ by the sets of labels of their children, the root gets labelled with $a$." (In fact, since $M$ correctly computes well-foundedness and countability this is a first-order definition in $M$, but that doesn't matter.) Now every element of $M\setminus im(i)$ must consist only of elements of $M\setminus im(i)$, but this contradicts well-foundedness.


Note that the ability to characterize well-foudnedness makes $\mathcal{L}\_{\omega\_1,\omega\_1}$ diverge wildly from the standard model-theoretic constraints we are used to for first-order logic or even $\mathcal{L}\_{\omega\_1,\omega}$. That said, the situation shouldn't be overstated: see [S. Friedman's *Model theory for $\mathcal{L}\_{\infty\omega\_1}$*](https://www.sciencedirect.com/science/article/pii/0168007284900125) for a discussion of what sort of "local" compactness phenomena (analogous to Barwise compactness) can still occur.


