print(fgh(2, 3)) # Output: 24 print(fgh('w', 2)) # Output: fgh(2,2) = 8
: a formal proof assistant that defines onote.fast_growing up to (\varepsilon_0). Because the definition is built on onote (a computable notation for ordinals), the function is fully computable, and one can evaluate small inputs like fast_growing_ε₀ 2 = 2048 .
The table reveals a powerful idea: each step in the hierarchy corresponds to a jump in the hyperoperation sequence. For instance, $f_2(n)$ doesn't just generate exponential numbers; it's the application of $f_1$, which leads to the formula $2^n n$. The numbers grow fast right from the start: fast growing hierarchy calculator
For any limit ordinal ( \lambda ), the calculator must return ( \lambda[n] ) for natural ( n ). Examples:
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. print(fgh(2, 3)) # Output: 24 print(fgh('w', 2)) #
No real-world computer will ever compute ( f_\omega_1^\textCK(10) ), because that would require solving the halting problem. But we can compute its shape —the skeleton of its growth. And in doing so, we touch something profound: the structure of infinity, made visible through the simple rule of repeated application.
. This wasn't just doing more work; it was changing the rules. At f sub omega And in doing so
The hierarchy is defined systematically starting from a basic successor function. For any non-negative integer , the functions are constructed using three fundamental rules: 1. The Base Case At the absolute bottom of the hierarchy ( ), the function simply increments the input by one. f0(n)=n+1f sub 0 of n equals n plus 1 2. The Successor Stage For any step where the index is a successor ordinal ( ), the function iterates the previous function level
At this stage, the calculator transcends standard arithmetic. roughly matches the Ackermann function ( ) and Knuth’s up-arrow notation ( Translating FGH to Other Large Number Notations
Logicians use ordinal analysis to measure the strength of formal systems. An FGH calculator helps visualize how fast a system’s provably total functions grow.
We can explore the definition of for higher ordinals like ωωomega raised to the omega power ϵ0epsilon sub 0 to see how the limit stage scales.