( f_\varepsilon_0(3) ) with Wainer fundamental sequences.
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 fast growing hierarchy calculator high quality
The output must be readable. A raw BigInt for $f_2(10)$ is readable. For $f_3(4)$, the output should be formatted as: ( f_\varepsilon_0(3) ) with Wainer fundamental sequences
: Such a calculator can serve as an educational tool, helping students understand the concepts of growth rates and computability. Core Definitions of the Fast-Growing Hierarchy The hierarchy
# Successor Ordinal if is_successor(alpha): # Try to derive closed form to avoid iteration stack overflow if alpha == 1: return x + x if alpha == 2: return x * (2**x) if alpha == 3: return tetration(x) # Symbolic Up-Arrow
Appendix: Minimal worked computation examples