For a class $W$:

Let $\mathcal{T}_0(W)=W$.

Let $\mathcal{T}_{\alpha+1}(W)=\{x\in V:x\subseteq\mathcal{T}_\alpha(W)\}\cup\mathcal{T}_\alpha(W)$.

Let $\mathcal{T}_{\beta}(W)=\bigcup_{\alpha\in\beta}\mathcal{T}_\alpha(W)$ for limit ordinals $\beta$.

Let $\mathrm{K}(W)=\min\{\alpha:\mathcal{T}_\alpha(W)=V\}$ (if such a minimum exists). The smaller this value is, the "closer" $W$ is to $V$.

With this definition, $\mathrm{K}(L)$ is guaranteed to be either a limit ordinal or $0$. Quite clearly $\mathrm{K}(L)=0\Leftrightarrow V=L$.

However, assuming $V\neq L$, is it possible to prove what $\mathrm{K}(L)$ is? Are there any models of ZFC in which it does not exist? (I.E. $M\models\forall\alpha\in\mathrm{Ord}(\mathcal{T}_\alpha(L)\neq V)$)

A couple facts that may help:

- $\mathcal{T}_\alpha(L)$ is transitive for every $\alpha$ (Proven by transfinite induction)
- $V_{\omega+\alpha}\in\mathcal{T}_\alpha(L)$ (Proven also by transfinite induction)
- $\mathrm{K}(L)\neq\alpha+1$ for any $\alpha$ (Transfinite induction)
- $V\setminus L$ is either the empty set or a proper class (Deduction using the fact that $L$ is transitive and a model of the axiom of pairing)