Let $L$ be the language of set theory, and for each countable ordinal $\kappa$, define $L_\kappa$ as follows: $L_0 = L$, and $L_{\alpha+1}$ is obtained by adding countably many new class constants $c_\phi$ to the signature of $L_\alpha$ to stand for each formula $\phi(x)$ of one free variable. Define $L_\alpha$ for $\alpha$ a limit ordinal to be the union of $L_\beta$ for $\beta < \alpha$. Given some set theory such as ZF, it is possible to add axioms for the new constants: let $T_0 = ZF$ and $T_{\alpha+1}$ be the theory got from $T_\alpha$ by adding $$x \in c_\phi \iff \phi(x)$$ as an axiom where $\phi(x)$ ranges over the countably many formulas in $L_\alpha$. Let $T_\alpha$ for $\alpha$ a limit ordinal be the union of $T_\beta$ for $\beta<\alpha$. This is ordinary expansion by definitions.

So, for each countable ordinal $\kappa$, $T_\kappa$ is countably axiomatizable. Define $L_\Omega$ and $T_\Omega$ as the union of $L_\kappa$ and $T_\kappa$ as $\kappa$ ranges over countable ordinals.

Is $T_\Omega$ countably axiomatizable?

What relation does MK have to $T_\Omega$, if any?

New question: is every formula of $L_\Omega$ equivalent to a formula of $L_1$ modulo $T_1$? (seems likely) please see: Is every formula of LΩ equivalent to a formula of L1 modulo T1?