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 (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 ${\mathrm{\aleph }}_1$.

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 h_α: 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 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 $\{{\mathscr{H}}_{\alpha }{\}}_{\alpha <\mu }$ is a family of numerical functions. For each ordinal Template:Math, a set ${\mathscr{H}}_{\alpha }$ is defined as the smallest class of functions containing Template:Math, zero, successor and projection functions, and closed under limited primitive recursion and limited substitution (similar to Grzegorczyk hierarchy).

Template:Harvp 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 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(n-1) ≤ H_ε0(n) ≤ f_ε0(n+1) for all n ≥ 1. Template:Harv

References

• Template:Citation
• Template:Citation. (In particular Section 12, pp. 59–64, "A Glimpse at Hierarchies of Fast and Slow Growing Functions".)
• Template:Citation.