Fast Growing Hierarchy Calculator High Quality [new] May 2026

Fast-Growing Hierarchy (FGH) Calculator – High-Quality Specification

1. Overview

The Fast-Growing Hierarchy (FGH) is a family of functions ( f_\alpha: \mathbbN \to \mathbbN ), indexed by ordinals ( \alpha ), that rigorously defines the concept of "very fast growth" in proof theory and computability theory.
A high-quality FGH calculator goes beyond simple recursion—it must handle limit ordinals, fundamental sequences, and large countable ordinals up to (and beyond) the Bachmann–Howard ordinal.

Example fundamental for ω

def fund_w(alpha, n): if alpha == 'ω': return n return alpha

Dependence on choice of fundamental sequences and ordinal notation means implementations must document conventions.
Numerical output quickly becomes unrepresentable; focus should be on asymptotic classification or symbolic forms.
For ordinals beyond commonly implemented notations (e.g., beyond ε0) more sophisticated ordinal collapsing functions are required, increasing complexity.

print(f(3, 3)) # 2↑↑3 = 16

Our calculator is designed to provide an accurate and efficient way to compute values within the Fast Growing Hierarchy. With a user-friendly interface, you can easily input values and explore the growth of numbers. Here are some key features:

1. Overview

Example fundamental for ω

def fund_w(alpha, n): if alpha == 'ω': return n return alpha

Dependence on choice of fundamental sequences and ordinal notation means implementations must document conventions.

Numerical output quickly becomes unrepresentable; focus should be on asymptotic classification or symbolic forms.

For ordinals beyond commonly implemented notations (e.g., beyond ε0) more sophisticated ordinal collapsing functions are required, increasing complexity.

print(f(3, 3)) # 2↑↑3 = 16