In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy, or a Schwichtenberg–Wainer hierarchy) is an ordinal-indexed family of rapidly increasing functions fα: N → N (where N is the set of natural numbers {0, 1, ...}, and α ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and computational complexity.
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Fast-growing hierarchy in the context of Power of 2
A power of two is a number of the form 2 where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent. In the fast-growing hierarchy, 2 is exactly equal to . In the Hardy hierarchy, 2 is exactly equal to .
Powers of two with non-negative exponents are integers: 2 = 1, 2 = 2, and 2 is two multiplied by itself n times. The first ten powers of 2 for non-negative values of n are:
View the full Wikipedia page for Power of 2Fast-growing hierarchy in the context of 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.
The Hardy hierarchy was 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 .
View the full Wikipedia page for Hardy hierarchy