Fast Growing Hierarchy Calculator [patched] Jun 2026

The most prominent online calculator is the . This JavaScript tool allows you to input a natural number (n) and a countable ordinal (\alpha) expressed in the normal form for the Extended Buchholz function, a powerful system of fundamental sequences that reaches far beyond the small Veblen ordinal. It is one of the few calculators that can handle ordinals beyond (\varepsilon_0). Another notable tool is the Ordinal Expander in JavaScript (ordex) , which is designed to expand ordinals and compute their fundamental sequences, which is the core operation for any FGH calculator.

The system is defined by three simple rules, starting with the most basic operation:

A calculator for this hierarchy allows users to input an ( ) and a natural number (

[ f_0(n) = n+1 ]

[ f_\alpha(n) = f_\alpha[n](n) ]

) . The calculator must interpret the ordinal, often written in Cantor Normal Form (e.g., 2. Symbolic Reduction

The hierarchy provides a framework to approach functions that grow too fast to be computed by any Turing machine, such as the Busy Beaver function ( ) or Rado's Sigma function . fast growing hierarchy calculator

Now wrap your mind around this: ( f_\omega+1(3) ) applies ( f_\omega ) three times, starting from 3. The first ( f_\omega(3) ) is that insane number. Then you apply ( f_\omega ) to that insane number. And then again. The result is barely within the realm of describable googology.

This isn't a tool but an incredible live example of the manual process. A community member provided a step-by-step expansion of $f_\omega^3(2)$, showing the nested iterations required to evaluate the function by hand. It's a fantastic demonstration of the underlying mechanics.

It is used to determine the termination of complex algorithms. If a proof's complexity can be mapped to an ordinal below ϵ0epsilon sub 0 , it can be proven sound within Peano arithmetic. The most prominent online calculator is the

and attempt to return the value (f_\alpha(n)).

The Fast-Growing Hierarchy (FGH) is a system of functions used in googology to name and categorize unimaginably large numbers. It outpaces standard notation like exponents or even Knuth's up-arrows by using transfinite ordinals. Core Functionality The hierarchy, denoted as , builds speed based on the index (the "ordinal") and the input : . This is simple successor logic. Successor Stage : . The function iterates itself Limit Stage : For limit ordinals (like ), we use a fundamental sequence: Notable Benchmarks As the index increases, the growth rate explodes. : Equal to . Linear growth. : Equal to . Exponential growth. : Comparable to Graham’s Number . It uses power towers.