The (FGH) is a family of functions ( f_\alpha : \mathbbN \to \mathbbN ) indexed by ordinals ( \alpha ). It is a central tool in proof theory and googology (the study of large numbers) for comparing the growth rates of functions and defining enormous numbers.
reached the first "limit ordinal." Here, the calculator didn't just add or multiply; it looked at the entire history of its growth and used that as its new starting point. The Moment
This isn't a tool but an incredible live example of the manual process. A community member provided a step-by-step expansion of $f_\omega^3(2)$, showing the nested iterations required to evaluate the function by hand. It's a fantastic demonstration of the underlying mechanics.
Use Wainer/Hardy style (commonly used in computability literature): fast growing hierarchy calculator
At the very bottom of the hierarchy, the function simply increments the input variable by one. It represents linear growth. 2. The Successor Step
| Index | Mathematical Formula | Approximate Growth Rate | | :--- | :--- | :--- | | $f_0(n)$ | $n+1$ | Addition | | $f_1(n)$ | $2n$ | Multiplication | | $f_2(n)$ | $2^n \cdot n$ | Exponential | | $f_3(n)$ | ≥ $2↑↑n$ | Tetration (Power Towers) | | $f_m(n)$ | ≥ $2↑^m-1n$ | Hyperoperation |
This guide explains fast-growing hierarchies (FGHs), how to compute values at small ordinals, practical strategies for a calculator implementation, algorithms and data structures, performance considerations, and examples. It assumes familiarity with ordinals up to ε0 and basic recursion theory; if not, the worked examples will still illustrate concrete cases. The (FGH) is a family of functions (
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The hierarchy is built using three fundamental rules based on the index f0(n)=n+1f sub 0 of n equals n plus 1
The is a mathematical framework used by googologists and theoretical computer scientists to define and compare functions that grow at staggering rates. It provides a standardized way to describe "ridiculously huge numbers" using ordinals to index the level of growth complexity. 🛠️ Core Definition The hierarchy consists of an indexed family of functions The Moment This isn't a tool but an
This is the successor function, the fundamental unit of growth. Successor Step
function, quickly reaching the boundaries of standard primitive recursive functions. : The index ϵ0epsilon sub 0
: This level matches the growth rate of the Ackermann function.
, which yields a massive integer over 5 million digits long. (using Knuth's up-arrow notation). : Entering the Transfinite fωf sub omega