Instead of calculating f₃(3) exactly, it calculates the number of digits or uses approximation techniques to describe the magnitude. For example, a calculator might inform you that
Therefore, almost all FGH calculators are —they work for very small (n) (typically 0, 1, or 2) but cannot produce the full output for (n \ge 3) except by symbolic manipulation.
A functional FGH calculator must handle symbolic ordinal arithmetic. 1. Ordinal Parsing The engine must recognize standard Cantor Normal Form. fast growing hierarchy calculator
times repeatedly. This creates an explosion of exponentiation and tetration.)
The Fast-Growing Hierarchy (FGH) is a system of functions used in googology to name and categorize unimaginably large numbers. It outpaces standard notation like exponents or even Knuth's up-arrows by using transfinite ordinals. Core Functionality The hierarchy, denoted as , builds speed based on the index (the "ordinal") and the input : . This is simple successor logic. Successor Stage : . The function iterates itself Limit Stage : For limit ordinals (like ), we use a fundamental sequence: Notable Benchmarks As the index increases, the growth rate explodes. : Equal to . Linear growth. : Equal to . Exponential growth. : Comparable to Graham’s Number . It uses power towers. Instead of calculating f₃(3) exactly, it calculates the
The hierarchy is built on three simple recursive rules that turn basic addition into "monster" functions:
For ordinals beyond a certain recursive bound, the question “Is this ordinal a limit ordinal?” can be undecidable. Real calculators restrict to and explicit fundamental sequences. This creates an explosion of exponentiation and tetration
This famously massive number is bounded tightly between in extended versions of the hierarchy, where represents the first transfinite ordinal. How an FGH Calculator Operates
