Hardy hierarchy

In computability theory, computational complexity theory and proof theory, the Hardy hierarchy, named after G. H. Hardy, is a hierarchy of sets of numerical functions generated from an ordinal-indexed family of functions h_α: N → N (where N is the set of natural numbers, {0, 1, ...}) called Hardy functions. It is related to the fast-growing hierarchy and slow-growing hierarchy.

The Hardy hierarchy was introduced by Stanley S. Wainer in 1972,^[1][2] but the idea of its definition comes from Hardy's 1904 paper,^[2][3] in which Hardy exhibits a set of reals with cardinality ${\displaystyle \aleph _{1}}$.