The system builds upon a base function, usually defined as a simple increment: f0(n)=n+1f sub 0 of n equals n plus 1 For any non-negative integer
Use Wainer/Hardy style (commonly used in computability literature):
The Fast-Growing Hierarchy is a indexed family of rapidly growing functions. It is typically denoted by is a non-negative integer and is an ordinal number. As the index fast growing hierarchy calculator
library to handle extremely large numbers and allows for powers of in calculations. : A general mathematical tool that includes an approximateFGH(x)
while True: user_input = input("Enter alpha (ordinal) and n (e.g., '2 3' for f_2(3)): ").strip() The system builds upon a base function, usually
fk+1(n)=fkn(n)f sub k plus 1 end-sub of n equals f sub k to the n-th power of n In this notation, means applying the function to the input times. For example, Growth Levels: From Addition to Graham's Number
An interactive tool that computes values of the fast-growing hierarchy ( f_\alpha(n) ) for user-provided ordinal ( \alpha ) (up to a reasonable limit, e.g., ( \Gamma_0 ) or less) and integer ( n ), with step-by-step expansion visualization. : A general mathematical tool that includes an
Even for relatively small inputs, the recursion depth and the size of the numbers become astronomical almost instantly. For instance, computing (f_\omega+1(3)) would involve iterating (f_\omega) three times, but (f_\omega(3)) itself already requires evaluating (f_3(3)), which is tetration. The result has millions of digits, and the intermediate steps require recursive function calls that quickly exceed the limits of any physical computer.
) increases, the rate of growth accelerates dramatically. The system starts with basic arithmetic and rapidly scales up to functions that outpace any standard computational model. The Core Rules of FGH
Therefore, architectural implementations do not compute the final digits. Instead, they to find bounds or convert between notations (such as converting Ackerman functions or Knuth up-arrows into their exact FGH equivalents). Comparing FGH to Other Large Number Notations