Working in $\mathcal L_{\omega_1, \omega_1}$, 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?

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

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

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

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

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

$ \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)\} $

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

Is this theory Complete?

Is this theory Categorical?

Working in $\mathcal L_{\omega_1, \omega_1}$, 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)\} $

Is this theory Complete?

Is this theory Categorical?

Working in $\mathcal L_{\omega_1, \omega_1}$, 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?