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 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, but the idea of its definition comes from Hardy's 1904 paper, in which Hardy exhibits a set of reals with cardinality.

Definition

Let μ be a large countable ordinal such that a fundamental sequence is assigned to every limit ordinal less than μ. The Hardy functions h_α: N → N, for α < μ, is then defined as follows:

if α is a limit ordinal.

Here α denotes the n^th element of the fundamental sequence assigned to the limit ordinal α. A standardized choice of fundamental sequence for all α ≤ ε₀ is described in the article on the fast-growing hierarchy.
The Hardy hierarchy is a family of numerical functions. For each ordinal, a set is defined as the smallest class of functions containing, zero, successor and projection functions, and closed under limited primitive recursion and limited substitution.
defines a modified Hardy hierarchy of functions by using the standard fundamental sequences, but with α in the third line of the above definition.

Relation to fast-growing hierarchy

The Wainer hierarchy of functions f_α and the Hardy hierarchy of functions H_α are related by f_α = H_ω^α for all α < ε₀. Thus, for any α < ε₀, H_α grows much more slowly than does f_α. However, the Hardy hierarchy "catches up" to the Wainer hierarchy at α = ε₀, such that f_ε₀ and H_ε₀ have the same growth rate, in the sense that f_ε₀ ≤ H_ε₀ ≤ f_ε₀ for all n ≥ 1.