Hardy hierarchy
In computability theory, computational complexity theory and proof theory, the Hardy hierarchy, named after G. H. Hardy, is an ordinal-indexed family of functions hα: N → N (where N is the set of natural numbers, {0, 1, ...}). It is related to the fast-growing hierarchy and slow-growing hierarchy. The hierarchy was first described in Hardy's 1904 paper, "A theorem concerning the infinite cardinal numbers".
Definition
Let μ be a large countable ordinal such that a fundamental sequence is assigned to every limit ordinal less than μ. The Hardy hierarchy of functions hα: N → N, for α < μ, is then defined as follows:
- 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.
Caicedo (2007) defines a modified Hardy hierarchy of functions 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)
References
- Hardy, G.H. (1904), "A theorem concerning the infinite cardinal numbers", Quarterly Journal of Mathematics, 35: 87–94
- Gallier, Jean H. (1991), "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF), Ann. Pure Appl. Logic, 53 (3): 199–260, doi:10.1016/0168-0072(91)90022-E, MR 1129778. (In particular Section 12, pp. 59–64, "A Glimpse at Hierarchies of Fast and Slow Growing Functions".)
- Caicedo, A. (2007), "Goodstein's function" (PDF), Revista Colombiana de Matemáticas, 41 (2): 381–391.