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 hα: N → N (where N is the set of natural numbers, {0, 1, ...}) called Hardy functions.
A standardized choice of fundamental sequence for all α ≤ ε0 is described in the article on the fast-growing hierarchy.
is defined as the smallest class of functions containing Hα, zero, successor and projection functions, and closed under limited primitive recursion and limited substitution[2] (similar to Grzegorczyk hierarchy).
Caicedo (2007) defines a modified Hardy hierarchy of functions
The Wainer hierarchy of functions fα and the Hardy hierarchy of functions Hα are related by fα = Hωα for all α < ε0.