The proposed system consists of three core modules: The , the Reduction Engine , and the Symbolic Output Formatter .
Calculating $f_3(4)$ results in a number with $19,729$ digits. Calculating $f_3(5)$ results in $10^10^10^10^10^4$. A numeric calculator would overflow memory instantly. Therefore, a high-quality calculator must use symbolic representation.
class Ordinal: """Represents an ordinal in Cantor normal form for α < ε₀.""" def (self, value): # value can be int, 'w', or tuple for ω^a * b + rest self.value = value
fλ(n)=fλ[n](n)f sub lambda of n equals f sub lambda open bracket n close bracket end-sub of n Growth Benchmarks As the index fast growing hierarchy calculator high quality
If you are looking for a , or if you want to understand the profound mathematics driving these systems, this comprehensive guide will break down the mechanics, the ordinal indexing, and how computational tools handle the uncomputable. What is the Fast-Growing Hierarchy?
A high-quality calculator does not hang. It provides:
that can display fundamental sequences and calculate both FGH and SGH (Slow-Growing Hierarchy) up to high ordinals like Rathjen's Quick Reference: How FGH Grows The proposed system consists of three core modules:
References (selective)
The boundary where simple recursive programming breaks down without optimization. fωf sub omega The Ackermann-style Diagonalization Grows faster than any primitive recursive function. fω+1f sub omega plus 1 end-sub Graham's Number Bounds Graham's Number ( ) sits snugly between fϵ0f sub epsilon sub 0 Goodstein Sequences / Kirby-Paris Hydra ϵ0epsilon sub 0 (Epsilon-Nought) is the limit of towers of
Because many users come to FGH to learn, a "high quality" tool includes: A numeric calculator would overflow memory instantly
f1(n)=f0n(n)=n+1+1+…+1=2nf sub 1 of n equals f sub 0 to the n-th power of n equals n plus 1 plus 1 plus … plus 1 equals 2 n At this level, the growth rate is strictly linear. Level: Exponential Growth Moving up, a total of
cannot be written out in base-10 digits, a high-quality calculator will output the result . It will reduce the calculation into other well-known large number formats, such as: Knuth's Up-Arrow Notation ( ↑up arrow Conway Chained Arrow Notation Steinhaus-Moser Notation Bowers Explicit Array Notation (BEAN) 4. Cross-Classification (The "Googology" Benchmark)
. The growth rate itself accelerates with every increase of the input variable. Anatomy of a High-Quality FGH Calculator
provides Python implementations of extremely fast-growing functions, including a helper function to view calculations step-by-step. Ordinal Calculator and Explorer : A community-developed Ordinal Explorer