Fast Growing Hierarchy Calculator !full!

is an ordinal number. Its recursive definition is remarkably simple, yet it leads to explosive growth:

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

To find the value of the next level, you nest (iterate) the previous function times, using the input as the starting value.

A major hurdle in building an FGH calculator is the speed at which values become uncomputable. fast growing hierarchy calculator

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The fast growing hierarchy calculator offers several advantages and applications:

[ \varepsilon_0[2] = \omega^\omega \quad\Rightarrow\quad f_\varepsilon_0(2) = f_\omega^\omega(2) ] is an ordinal number

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

behind these levels, or should we continue Cali's journey into the Uncountable Ordinals

causes standard computer memory to overflow instantly due to deep recursion. A major hurdle in building an FGH calculator

Because these definitions are purely recursive and involve only natural numbers and ordinals, the functions are , at least in principle. In fact, the concept is so fundamental that the OEIS entry A275000 lists the main diagonal (F[n]_n(2)) of a related “fast‑iteration” function, with terms like 2, 4, 18, 590295810358705651712, … and the next term already too large to include.

The definition of the FGH is not unique—it depends on the chosen system of fundamental sequences. For ordinals beyond (\varepsilon_0) (e.g., the Veblen hierarchy, the Feferman–Schütte ordinal (\Gamma_0)), no universally accepted “standard” sequences exist, and the definition becomes more complex. An FGH calculator that aims to handle large ordinals must therefore allow the user to select a fundamental‑sequence system or implement a system like the or ordinal collapsing functions .