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 hα: 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.

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 $$\aleph_1$$.

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


 * $$ H_0(n) = n,$$
 * $$ H_{\alpha+1}(n) = H_\alpha(n + 1),$$
 * $$ H_{\alpha}(n) = H_{\alpha[n]}(n) $$ 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 $$\{\mathcal{H}_\alpha\}_{\alpha<\mu}$$ is a family of numerical functions. For each ordinal $α$, a set $$\mathcal{H}_\alpha$$ is defined as the smallest class of functions containing $H_{α}$, zero, successor and projection functions, and closed under limited primitive recursion and limited substitution (similar to Grzegorczyk hierarchy).

defines a modified Hardy hierarchy of functions $$H_\alpha$$ 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 fε 0 and Hε 0 have the same growth rate, in the sense that fε 0 (n-1) ≤ Hε 0 (n) ≤ fε 0 (n+1) for all n ≥ 1.