A high‑quality FGH calculator can be extended:
Use recursion with caching of ( f_\alpha(n) ) for small ( \alpha, n ).
from functools import lru_cache
@lru_cache(maxsize=None) def f(alpha, n): if n == 0: return 0 # or 1, depending on convention if alpha == 0: return n + 1 if is_successor(alpha): pred = predecessor(alpha) # iterate n times result = n for _ in range(n): result = f(pred, result) return result else: # limit return f(fund(alpha, n), n)fast growing hierarchy calculator high quality
Problem: This recursion is extremely deep for moderate n (e.g., ( f_\omega+1(3) ) already huge).
So high‑quality calculators must: A high‑quality FGH calculator can be extended:
def fgh(alpha, n, limit_ordinal_fundamental=None):
"""
Compute f_alpha(n) with custom fundamental sequences.
Args:
alpha: int or callable for limit ordinals returning alpha[n]
n: int >= 0
limit_ordinal_fundamental: function(alpha, n) -> alpha_n
"""
if alpha == 0:
return n + 1
if isinstance(alpha, int): # successor
result = n
for _ in range(n):
result = fgh(alpha - 1, result, limit_ordinal_fundamental)
return result
# limit ordinal
if limit_ordinal_fundamental:
alpha_n = limit_ordinal_fundamental(alpha, n)
return fgh(alpha_n, n, limit_ordinal_fundamental)
raise ValueError(f"No fundamental sequence for alpha")
If you are a developer wanting to create the ultimate FGH calculator, or a user hoping to locate one, here is the blueprint.
For inputs like ( f_\omega+1(4) ), the output is astronomically large (beyond power towers). A high-quality calculator does not attempt to print 10^10^... digits. Instead, it outputs: Use recursion with caching of ( f_\alpha(n) )
Replace recursion with a stack machine to avoid recursion limits:
Input: (alpha, n)
Stack = [(alpha, n)]
While stack not empty:
Pop (a, m)
if m == 0 → push result
else reduce a to a[m-1] …
This module handles the transfinite ordinals ($\omega, \omega+1, \omega \cdot 2, \omega^2, \epsilon_0$).
Requirements:
Fundamental Sequence Logic: For limit ordinals, the system must determine $\lambda[n]$.
What does "high quality" actually mean in this context? Let us break down the indispensable features.