Fast Growing Hierarchy Calculator

Show small numeric checks (calculator can output exact for these small α,n).

Imagine an open-source web app with:

This would be the Large Number Enthusiast’s slide rule—a window into the abyss of fast-growing functions.

Search online for “FGH calculator,” and you’ll find toy scripts that handle ( f_\alpha(n) ) for ( \alpha < \omega^2 ) and ( n < 5 ). A full-featured one is a beast. fast growing hierarchy calculator

By the time you reach f_ω(n), you are at the limit of primitive recursive functions (Ackermann function territory). By f_ε₀(n), you surpass the proof-theoretic strength of Peano arithmetic.

The core problem: Performing ( f_3(4) ) by hand is tedious. Performing ( f_ω+1(3) ) without a calculator is virtually impossible for a human. This is why we need a Fast Growing Hierarchy calculator.

For a given f_α(n):

Let’s trace a tiny example to appreciate the explosion:

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.

A proper FGH calculator would let you explore this madness with a few keystrokes. Show small numeric checks (calculator can output exact

With standard fundamental sequences:

[ f_\omega(3) = f_3(3) ] where ( f_3(3) ) is already enormous (much larger than ( 2 \uparrow\uparrow 3 )).