= Maurey hierarchy of finite sets
{c}
{title2=$\mathcal M_\alpha$}
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>.
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