As the index (the subscript) increases, the growth rate of the function accelerates dramatically. The hierarchy allows mathematicians to categorize large numbers by mapping them to specific levels of ordinal complexity. Core Mechanics and Definition
The text above provides the complete logic and code for a Fast Growing Hierarchy calculator. Due to the nature of the function, a standard numeric calculator can only function for $\alpha < 3$. Beyond that point, the "calculator" must switch to symbolic logic to describe the operations rather than the final number.
As the index (the subscript) of the function increases, the rate of growth accelerates at a pace that quickly eclipses any function found in traditional physics or standard calculus. The Fundamental Rules of FGH fast growing hierarchy calculator
The fast-growing hierarchy is a collection of functions, each of which grows faster than the previous one. It's a way to classify functions based on their growth rates. The hierarchy is often used to demonstrate the limits of computability and to study the complexity of mathematical functions.
The FGH is used to classify the provably total functions of various formal systems. For example, the functions that are provably total in Peano arithmetic are exactly those that are bounded by (f_\varepsilon_0) in the Wainer hierarchy. By implementing the hierarchy, one can obtain concrete examples of such functions. As the index (the subscript) increases, the growth
if alpha == 'w': return f"prefix -> f_n(n) ..."
If $\alpha$ is a limit ordinal (like $\omega$ or $\omega \times 2$), we use fundamental sequences. $$f_\alpha(n) = f_\alpha[n](n)$$ Translation for the calculator: Find the $n$-th element in the fundamental sequence of $\alpha$ and evaluate that function. Due to the nature of the function, a
“The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. But we can still talk about it sensibly—especially when we have a calculator.” — Paraphrasing Hilbert, with apologies.