Suppose $\mathbb{V} = \mathbb{L}$ and there is a countable transitive model $\mathbb{M}$ of $ZFC$.

Let $\rho$ be the $\mathbb{L}$-rank, i.e. for all $a \in \mathbb{V}$, $\rho(a) = $the least $\alpha$ with $a \in L_{\alpha+1}$.

Define a pre-order $<'$ on $M$ by $a <' b$ iff $\rho(a) < \rho(b)$.

Then my first question is: under what circumstances is $<'$ first-order definable over $\mathbb{M}$?

My second question is: supposing $(\mathbb{V} \not= \mathbb{L})^\mathbb{M}$, and given $a, b \in M \backslash \mathbb{L}^\mathbb{M}$, when does $\mathbb{M}$ "know" that $a <' b$?

Formally, when is there a formula $\phi(x, y)$ (without parameters) such that $\mathbb{M} \models \phi(a, b)$ and for all $a', b'$ with $\mathbb{M} \models \phi(a', b')$, we have $a' <' b'$?