Fast Growing Hierarchy Calculator !exclusive! Jun 2026

causes standard computer memory to overflow instantly due to deep recursion.

Building or using an FGH calculator requires understanding how different mathematical notation systems map onto specific ordinals. A robust calculator processes inputs across three primary tiers of infinity. 1. The Finite Levels (

This site shows how programmers try to implement extremely fast-growing FGH functions in as few characters as possible. For instance, one user's program results in $f_\textTFBO+1(3)$, where TFBO is the Takeuti-Feferman-Buchholz ordinal, a vastly powerful function. fast growing hierarchy calculator

Eventually we obtain

fα+1(n)=fαn(n)f sub alpha plus 1 end-sub of n equals f sub alpha to the n-th power of n (Note: means applying the function fαf sub alpha to the input , then applying it again to the result, repeated times). 3. The Limit Ordinal Case When the index reaches a limit ordinal (like causes standard computer memory to overflow instantly due

fβ+1(n)=fβn(n)f sub beta plus 1 end-sub of n equals f sub beta to the n-th power of n This means you apply the previous function fβf sub beta to the input times. For example,

It provides a universal "yardstick." Instead of debating whether Graham's Number or TREE(3) is bigger, mathematicians determine their positions on the hierarchy. Eventually we obtain fα+1(n)=fαn(n)f sub alpha plus 1

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

For a basic calculator, we implement these as predefined logic cases.

Given a fixed system of for limit ordinals, the hierarchy is defined recursively as follows:

The hierarchy helps mathematicians determine the strength of logical frameworks. For example, some mathematical theorems (like Goodstein's Theorem or the Kirby-Paris Hydra Game) produce sequences that are guaranteed to terminate, but the proof of their termination requires growth rates indexed by transfinite ordinals found deep within the Fast-Growing Hierarchy.