Fast Growing Hierarchy Calculator Jun 2026

The fast growing hierarchy calculator offers several advantages and applications:

A common choice is : ( \alpha = \omega^\beta_1 \cdot c_1 + \dots + \omega^\beta_k \cdot c_k ) with ( \beta_1 > \dots > \beta_k ).

Building a computer program to evaluate FGH values presents a unique paradox: a computer cannot output the digits of the numbers being calculated because the numbers are too large to fit within the physical data storage limits of the universe.

However, users should be aware of the calculator's limitations, particularly with regards to scalability and custom function support. fast growing hierarchy calculator

Repeated exponentiation leads to tetration, or power towers ( in Knuth's up-arrow notation). , which yields a massive power tower of 2s. — The Ackermann Rate The first limit ordinal is

— Linear Growth: Iterating addition yields multiplication.

print(f"\nCalculating f_alpha_val(n_in)...") Repeated exponentiation leads to tetration, or power towers

Implementing FGH efficiently stresses recursion, lazy evaluation, and memory management. Competing to compute ( f_\omega+1(5) ) symbolically is a brutal test for Haskell, Scheme, or Rust.

def f(alpha, n, limits): # limits: max_steps, max_bits key = (alpha.serialize(), n) if key in cache: return cache[key] if alpha.is_zero(): return n+1 if alpha.is_successor(): beta = alpha.predecessor() # compute iterate of f_beta, repeated n times starting at n val = iterate(lambda x: f(beta, x, limits), n, n, limits) cache[key] = val; return val # alpha is limit beta = alpha.fundamental(n) val = f(beta, n, limits) cache[key] = val; return val

If you want to dive deeper, explore the Wikipedia article on the fast‑growing hierarchy for the formal definitions, the Googology Wiki for comparisons to other notations, or simply run one of the GitHub implementations mentioned above and see how far your computer can go before the numbers become too large to print. print(f"\nCalculating f_alpha_val(n_in)

f_ε_0(2) with ε_0[n] = ω↑↑(n+1)

The fast-growing hierarchy consists of several functions, each denoted by a Greek letter (usually ω or Ω). The functions are defined recursively, with each function growing faster than the previous one. Here are the first few functions in the hierarchy:

: Pentational growth. This level easily generates , which sits comfortably below Transfinite Ordinals: Entering the Infinite When the index reaches