Rank of a well-founded tree ($r(t)=\sup \{r(t^{⌢}x)+1:t^{⌢}x\in T\}$)

The rank of a terminal node is zero; the rank of any node is the supremum of the successor ranks of its immediate extensions. The rank of the empty node gives one convention for tree height. Every countable well-founded tree has countable rank. Changing between empty-root rank and successive leaf-deletion order changes harmless endpoint conventions.

This hierarchy measures the complexity of successive finite selections. Start with ${\mathcal{M}}_0=\{\varnothing \}$. At a successor stage, retain the previous family and allow one smaller integer to be prepended. At a countable limit $\lambda$, choose positive ordinals ${\lambda }_j\uparrow \lambda$ and admit sets in ${\mathcal{M}}_{{\lambda }_j}$ whose minimum is at least $j$. Each family is hereditary and spreading, has no infinite branch, and its extension-tree rank on every infinite subset is at least $\alpha$. These properties make the hierarchy useful for transfinite constructions of unconditional basic sequences.