# 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 hαN → 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. Hardy hierarchy is 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 $\displaystyle{ \aleph_1 }$.

## 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 hαN → N, for α < μ, is then defined as follows:

• $\displaystyle{ h_0(n) = n, }$
• $\displaystyle{ h_{\alpha+1}(n) = h_\alpha(n + 1), }$
• $\displaystyle{ h_\alpha(n) = h_{\alpha[n]}(n) }$ if α is a limit ordinal.

Here α[n] denotes the nth 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 $\displaystyle{ \{\mathcal{H}_\alpha\}_{\alpha\lt \mu} }$ is a family of numerical functions. For each ordinal α, a set $\displaystyle{ \mathcal{H}_\alpha }$ is defined as the smallest class of functions containing hα, zero, successor and projection functions, and closed under limited primitive recursion and limited substitution (similar to Grzegorczyk hierarchy).

Caicedo (2007) defines a modified Hardy hierarchy of functions $\displaystyle{ H_\alpha }$ by using the standard fundamental sequences, but with α[n+1] (instead of α[n]) in the third line of the above definition.

## Relation to fast-growing hierarchy

The Wainer hierarchy of functions fα and the Hardy hierarchy of functions hα are related by fα = hωα for all α < ε0. Thus, for any α < ε0, hα grows much more slowly than does fα. However, the Hardy hierarchy "catches up" to the Wainer hierarchy at α = ε0, such that fε0 and hε0 have the same growth rate, in the sense that fε0(n-1) ≤ hε0(n) ≤ fε0(n+1) for all n ≥ 1. (Gallier 1991)