This level surpasses standard exponential notation. It creates towers of exponents, roughly equivalent to Knuth's up-arrow notation ( ). Even for small inputs like , the output is an astronomical tower of powers. Beyond Level 3: The Truly Massive : Comparable to pentation (
The primary use of FGH calculators is in googology—the study of large numbers. Enthusiasts assign a “FGH level” to a new function or notation (e.g., “this function is at level (f_\varepsilon_0)”) to indicate its growth rate. A calculator that can evaluate small cases helps verify these assignments.
Instead, an FGH calculator is best implemented as a . It takes a function definition and an input, and it applies the recursive rules until the expression is simplified or evaluated.
Below is a complete guide and a functional code implementation for an FGH Calculator. fast growing hierarchy calculator
The Fast-Growing Hierarchy (FGH) is a mathematical framework used to classify and compare rapidly growing functions. It provides a structured way to understand immense numbers that dwarf standard notation systems like scientific notation or even Tetration.
A standard calculator stores numbers as fixed floating-point values. An FGH calculator operates as a . Instead of storing the computed value, it stores the recipe for the number. 1. The Three Fundamental Rules
To put its power into perspective, standard arithmetic operations like addition, multiplication, and exponentiation represent only the absolute lowest rungs of this infinite ladder. The hierarchy builds upon itself using three core rules to define how functions escalate at different levels. The Three Core Rules of FGH This level surpasses standard exponential notation
Ordinals are not integers. The calculator must support:
— Linear Growth: Iterating addition yields multiplication.
(for a limit ordinal (\alpha)):
: A specialized tool for calculating FGH expressions using the Extended Buchholz Function . It allows you to input natural numbers and countable ordinals in normal form to see the resulting growth. Hardy Hierarchy Calculator
“The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. But we can still talk about it sensibly—especially when we have a calculator.” — Paraphrasing Hilbert, with apologies.