) is difficult, but it is tiny compared to Skewes' number, Graham's number, or TREE(3).
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
Beyond being a tool for googologists, the FGH has profound implications in mathematical logic and proof theory. It provides a way to measure the strength of formal systems: the smallest ordinal (\alpha) such that the function (f_\alpha) is not provably total in a given system is a measure of that system's proof-theoretic strength. For example, the well-ordering of (\varepsilon_0) is provable in Peano arithmetic, and the function (f_{\varepsilon_0}) corresponds to the growth rate of Goodstein sequences.
Derived from Kruskal's tree theorem; vastly outgrows Graham's number. Far beyond FGH fast growing hierarchy calculator
This public link is valid for 7 days and shares a thread, including any personal information you added. This link or copies made by others cannot be deleted. If you share with third parties, their policies apply. Can’t copy the link right now. Try again later.
It translates the FGH expression into a known large number notation (Conway chained arrows, BEAF, or TREE sequence comparisons).
The hierarchy is generally defined for an ordinal α < μ (where μ is a defined ordinal limit, often ε₀ or higher): ) is difficult, but it is tiny compared
In computational complexity, the FGH helps classify computable functions by their rate of growth and algorithmic complexity. The Wainer hierarchy, in particular, is intimately related to the , which classifies the primitive recursive functions.
Writing an FGH calculator is a rite of passage for functional programmers. It forces you to master recursion, memoization, and lazy evaluation. Handling ( f_{ω^{ω}}(n) ) requires implementing ordinal addition and multiplication.
, which are the "instructions" for breaking down complex ordinals like epsilon sub 0 Mathematics Stack Exchange Golf the fast growing hierarchy - Code Golf Stack Exchange It provides a way to measure the strength
Given a fixed system of for limit ordinals, the hierarchy is defined recursively as follows:
The "Fast Growing Hierarchy" (FGH) is a framework used in googology (the study of large numbers) to compare the growth rates of functions. Because the values produced by this hierarchy quickly become too large for standard computer arithmetic (even exceeding the estimated number of atoms in the universe within the first few steps), a "calculator" in the traditional sense (input number -> output number) is impossible for higher levels.
A Fast-Growing Hierarchy calculator changes how we view mathematical infinity. Rather than treating massive values as abstract concepts, it organizes them into a strict, verifiable structure. By breaking down complex notations into foundational rules, these tools allow mathematicians and enthusiasts to map the farthest reaches of numerical growth.