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
, which represents the "limit" of all natural numbers), the function "diagonalizes" by choosing a level from the hierarchy based on the input .
Safeguards:
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. fast growing hierarchy calculator
, the function is defined by iterating the previous function times on the input Limit Step
) is difficult, but it is tiny compared to Skewes' number, Graham's number, or TREE(3).
Below is a working designed to handle the hierarchy up to $\varepsilon_0$. It utilizes JavaScript’s native BigInt to handle large integers. A standard calculator stores numbers as fixed floating-point
if __name__ == "__main__": main()
Derived from Kruskal's tree theorem, TREE(3) is incomprehensibly larger than Graham's number. Its growth rate corresponds to the ordinal Γ0cap gamma sub 0 (the Feferman–Schütte ordinal), placing it near
These online calculators can be used to explore the properties of the fast growing hierarchy functions and to gain insight into their growth rates. It assumes familiarity with ordinals up to ε0
: a Python implementation of the Wainer hierarchy that tries to compute the functions strictly according to the recursive definition. The author notes that “for almost all input values this function will never return any value as the runtime will be far too long,” but the code is intended to be a faithful computational model of the concept.
The Fast-Growing Hierarchy is more than an abstract mathematical concept; it is the definitive language for describing and comparing the most extreme growth rates in all of mathematics. While a simple web calculator for the FGH is elusive, the resources listed here—spanning live calculations, open-source code, and advanced JavaScript libraries—provide a powerful and comprehensive toolkit for anyone ready to explore this breathtaking mathematical frontier.