If one replaces the cofinality $\omega$ requirement by an arbitrary cofinality (and if one also insists that $\alpha_m\lt\alpha$, as seems intended), then the construction is the same as that of the Cantor-Bendixon derivative. This is an idea that applies in any topological space, and ordinals are topological spaces under the order topology.

One begins with $X_0$ as the whole space, and then defines $X_{\alpha+1}$ to be all the limit points of $X_\alpha$, and for limit ordinals $\lambda$, one defines $X_\lambda=\bigcap_{\alpha\lt\lambda}X_\alpha$. Thus, the process proceeds by throwing out all the isolated points, and then throwing out the isolated points among the remaining points, and so on, iterating transfinitely, taking intersections at limit stages. The Cantor-Bendixon rank of a point $x$ is the first ordinal stage $\alpha$ at which it becomes isolated among the remaining points at that stage; in other words, the least $\alpha$, such that $x\notin L_{\alpha+1}$.

If the initial space is all the ordinals, then the Cantor-Bendixon construction is identical to the $L_\beta$ construction (corrected as indicated above). In particular, the $\beta$-limit ordinals (as corrected) are precisely the ordinals having Cantor-Bendixon rank at least $\beta$.

If one replaces the cofinality $\omega$ requirement by an arbitrary cofinality (and if one also insists that $\alpha_m\lt\alpha$, as seems intended), then the construction is the same as that of the Cantor–Bendixson derivative. This is an idea that applies in any topological space, and ordinals are topological spaces under the order topology.

One begins with $X_0$ as the whole space, and then defines $X_{\alpha+1}$ to be all the limit points of $X_\alpha$, and for limit ordinals $\lambda$, one defines $X_\lambda=\bigcap_{\alpha\lt\lambda}X_\alpha$. Thus, the process proceeds by throwing out all the isolated points, and then throwing out the isolated points among the remaining points, and so on, iterating transfinitely, taking intersections at limit stages. The Cantor–Bendixson rank of a point $x$ is the first ordinal stage $\alpha$ at which it becomes isolated among the remaining points at that stage; in other words, the least $\alpha$, such that $x\notin L_{\alpha+1}$.

If the initial space is all the ordinals, then the Cantor–Bendixson construction is identical to the $L_\beta$ construction (corrected as indicated above). In particular, the $\beta$-limit ordinals (as corrected) are precisely the ordinals having Cantor–Bendixson rank at least $\beta$.

If one replaces the cofinality $\omega$ requirement by an arbitrary cofinality (and if one also insists that $\alpha_m\lt\alpha$, as seems intended), then the construction is the same as that of the Cantor-Bendixon derivative. This is an idea that applies in any topological space, and ordinals are topological spaces under the order topology.

One begins with $X_0$ as the whole space, and then defines $X_{\alpha+1}$ to be all the limit points of $X_\alpha$, and for limit ordinals $\lambda$, one defines $X_\lambda=\bigcap_{\alpha\lt\lambda}X_\alpha$. Thus, the process proceeds by throwing out all the isolated points, and then iterating this, taking intersections at limit stages. The Cantor-Bendixon rank of a point $x$ is the first ordinal $\alpha$ such that $x\notin L_{\alpha+1}$, that is, the first stage at which it becomes isolated.

If the initial space is all the ordinals, then the Cantor-Bendixon construction is identical to the $L_\beta$ construction (corrected as indicated above). In particular, the $\beta$-limit ordinals (as corrected) are precisely the ordinals having Cantor-Bendixon rank at least $\beta$.

If one replaces the cofinality $\omega$ requirement by an arbitrary cofinality (and if one also insists that $\alpha_m\lt\alpha$, as seems intended), then the construction is the same as that of the Cantor-Bendixon derivative. This is an idea that applies in any topological space, and ordinals are topological spaces under the order topology.

One begins with $X_0$ as the whole space, and then defines $X_{\alpha+1}$ to be all the limit points of $X_\alpha$, and for limit ordinals $\lambda$, one defines $X_\lambda=\bigcap_{\alpha\lt\lambda}X_\alpha$. Thus, the process proceeds by throwing out all the isolated points, and then throwing out the isolated points among the remaining points, and so on, iterating transfinitely, taking intersections at limit stages. The Cantor-Bendixon rank of a point $x$ is the first ordinal stage $\alpha$ at which it becomes isolated among the remaining points at that stage; in other words, the least $\alpha$, such that $x\notin L_{\alpha+1}$.

If the initial space is all the ordinals, then the Cantor-Bendixon construction is identical to the $L_\beta$ construction (corrected as indicated above). In particular, the $\beta$-limit ordinals (as corrected) are precisely the ordinals having Cantor-Bendixon rank at least $\beta$.