Morley rank

Definition

Let $M$ be an $L$-structure that is $\aleph_0$-saturated, and let $\varphi(\bar{x})$ be an $L$-formula with parameters from $M$. Write $\varphi(M)$ for the definable set of tuples of $M$ that satisfy $\varphi$. For each ordinal $\alpha$, the relation $\mathrm{RM}(\varphi) \ge \alpha$ is defined by induction on $\alpha$:

1. $\mathrm{RM}(\varphi) \ge 0$ if $\varphi(M)$ is nonempty;
2. $\mathrm{RM}(\varphi) \ge \alpha + 1$ if there are formulas $\psi_0, \psi_1, \dots$ with parameters from $M$ whose sets $\psi_i(M)$ are pairwise disjoint subsets of $\varphi(M)$ with $\mathrm{RM}(\psi_i) \ge \alpha$;
3. for a limit ordinal $\alpha$, $\mathrm{RM}(\varphi) \ge \alpha$ if $\mathrm{RM}(\varphi) \ge \beta$ for every $\beta < \alpha$.

The Morley rank $\mathrm{RM}(\varphi)$ is $-1$ if $\varphi(M)$ is empty. Otherwise it is the largest ordinal $\alpha$ with $\mathrm{RM}(\varphi) \ge \alpha$ if one exists, and $\infty$ if $\mathrm{RM}(\varphi) \ge \alpha$ for every ordinal $\alpha$. For an arbitrary $L$-structure $N$ and a formula with parameters from $N$, the Morley rank is computed in an $\aleph_0$-saturated elementary extension of $N$. It does not depend on the choice of the extension.