Fast-Growing Hierarchy (FGH) is an ordinal-indexed system of functions used by mathematicians and "googologists" to classify and generate incredibly large numbers. While a "calculator" in the traditional sense is often impossible for high-level ordinals due to the sheer scale of the outputs, various online tools and algorithms have been developed to explore these functions and their underlying ordinal structures. Core Definitions of the Fast-Growing Hierarchy The hierarchy consists of a family of functions defined by three recursive rules: Successorship (Base Case): Successor Ordinal: (Applying the previous function Limit Ordinal: (Using the th term of a "fundamental sequence" assigned to Growth benchmarks and levels As the index increases, the growth rate of f sub alpha : Simple doubling. : Eventually dominates standard exponential functions. : Comparable to tetration ( ) and the standard Ackermann function : Grows roughly as fast as , outstripping any function with a finite index. : Often used to approximate Graham's Number Allam's Numbers - The Fast Growing Hierarchy
Final recommendation:
Start with a Python class supporting Cantor normal form, add a fundamental method, and cap n ≤ 4 for practical use. For large ordinals, output the growth rate symbolically rather than computing exact integers. fast growing hierarchy calculator high quality
The hierarchy is defined by three rules that describe how to move from simple counting to functions that grow faster than any computable function: Buchholz function Fast-Growing Hierarchy (FGH) is an ordinal-indexed system of
Common convention (Wainer hierarchy):
For ( \alpha < \varepsilon_0 ): F₀(x) = x + 1
F₁(x) = F₀(F₀(
def fgh(alpha, n, limit_ordinal_fundamental=None):
"""
Compute f_alpha(n) with custom fundamental sequences.
Args:
alpha: int or callable for limit ordinals returning alpha[n]
n: int >= 0
limit_ordinal_fundamental: function(alpha, n) -> alpha_n
"""
if alpha == 0:
return n + 1
if isinstance(alpha, int): # successor
result = n
for _ in range(n):
result = fgh(alpha - 1, result, limit_ordinal_fundamental)
return result
# limit ordinal
if limit_ordinal_fundamental:
alpha_n = limit_ordinal_fundamental(alpha, n)
return fgh(alpha_n, n, limit_ordinal_fundamental)
raise ValueError(f"No fundamental sequence for alpha")
Given the rapid growth rate of these functions, manual computation is impractical, and a reliable calculator is essential for exploring the fast-growing hierarchy. Our calculator is designed to provide accurate and efficient computation of these functions, allowing researchers and enthusiasts to: