| 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 |
Beyond the finite ordinals, the FGH uses the fundamental sequences of limit ordinals like (\omega), (\varepsilon_0), and far beyond to produce functions that dwarf even the most powerful combinatorial functions. For example, the famous Goodstein sequences, which are not provably total in Peano arithmetic, have growth rates comparable to (f_\varepsilon_0) in a fast-growing hierarchy.
If the index $\alpha$ is a successor ordinal (e.g., $1, 2, 3$): $$f_\alpha+1(n) = f_\alpha^n(n)$$ Note: The superscript denotes iteration. $f_\alpha^n(n)$ means apply $f_\alpha$ to $n$, then apply it to the result, repeating $n$ times. fast growing hierarchy calculator
While it may seem like pure mathematical recreation, the Fast-Growing Hierarchy is an indispensable tool in theoretical computer science and mathematical logic.
Because the actual numbers cannot be stored in standard computer memory, the calculator outputs the number using alternative notations (e.g., Knuth's Up-Arrow, Conway Chained Arrow, or Bowers Exploding Array Notation). The Challenge of Computability | Index | Mathematical Formula | Approximate Growth
: Achieves growth rates comparable to tetration and Graham's Number once reaches slightly higher levels like . 3. The Role of the Calculator
It breaks down limit ordinals using pre-defined rules, such as $f_\alpha^n(n)$ means apply $f_\alpha$ to $n$, then apply
While a dedicated online tool is rare, several powerful programming libraries and conceptual calculators exist. These are your best resources.
fα(n)=fα[n](n)f sub alpha of n equals f sub alpha open bracket n close bracket end-sub of n 2. Levels of Growth As the index