Working in [$\mathcal L_{\omega_1, \omega_1}$][1], add symbol $=$ with its axioms; add symbol $\in$ and axiomatize:

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


$\textbf{Foundation: }  (\forall v_n)_{n \in \omega} \, \exists x: \bigvee_{n \in \omega} (x=v_n) \land \bigwedge_{n \in \omega} (v_n \not \in x)$

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

$\textbf{Construction: } \\(\forall v_n)_{n \in \omega} \, \exists x : x=\{y \mid  y \neq v_0\land \bigvee_{n \in \omega} ( y=v_n)\}$


 $ \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)\} $

$\textbf {Multiplicity: } \exists x \exists y: x \neq y$.




> Is this theory Complete?

> Is this theory Categorical?
 


  [1]: https://plato.stanford.edu/entries/logic-infinitary/