Hardy hierarchy

From HandWiki

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,[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 [math]\displaystyle{ \aleph_1 }[/math].


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:

  • [math]\displaystyle{ h_0(n) = n, }[/math]
  • [math]\displaystyle{ h_{\alpha+1}(n) = h_\alpha(n + 1), }[/math]
  • [math]\displaystyle{ h_\alpha(n) = h_{\alpha[n]}(n) }[/math] 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 [math]\displaystyle{ \{\mathcal{H}_\alpha\}_{\alpha\lt \mu} }[/math] is a family of numerical functions. For each ordinal α, a set [math]\displaystyle{ \mathcal{H}_\alpha }[/math] 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 [math]\displaystyle{ H_\alpha }[/math] 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)


  1. Fairtlough, Matt; Wainer, Stanley S. (1998). "Chapter III - Hierarchies of Provably Recursive Functions". Handbook of Proof Theory. 137. Elsevier. pp. 149–207. doi:10.1016/S0049-237X(98)80018-9. ISBN 9780444898401. 
  2. 2.0 2.1 2.2 Wainer, S. S. (1972). "Ordinal recursion, and a refinement of the extended Grzegorczyk hierarchy" (in en). The Journal of Symbolic Logic 37 (2): 281–292. doi:10.2307/2272973. ISSN 0022-4812. https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/ordinal-recursion-and-a-refinement-of-the-extended-grzegorczyk-hierarchy/C47A374233B902E669249E46945163F1. 
  3. Hardy, G.H. (1904). "A THEOREM CONCERNING THE INFINITE CANONICAL NUMBERS". Quarterly Journal of Mathematics 35: 87–94.