Fast Growing Hierarchy | Calculator //top\\

This script acts as a symbolic calculator. It can compute values for $f_0, f_1, f_2, \dots, f_\omega$. Note that $f_3(3)$ already yields a number with over 3 trillion digits. This program will stop if the number becomes too large to store in memory, but it will print the reduction steps for any valid input.

It translates the FGH expression into a known large number notation (Conway chained arrows, BEAF, or TREE sequence comparisons).

As you can see, these functions grow extremely rapidly. The function $f_0(n)$ is simply $n + 1$, but $f_1(n)$ grows to $2n + 1$, $f_2(n)$ grows to $2^2n + 1 + 1$, and $f_3(n)$ grows to $2^2^2n + 1 + 1 + 1$. This rapid growth makes it difficult to compute these functions by hand, which is where the fast growing hierarchy calculator comes in.

In most of our daily lives, numbers are tame. They count apples, measure distances, or track bank balances. Even a "big number" like a trillion is merely a fly on the wall of the mathematical universe. fast growing hierarchy calculator

: The logic became so complex that Cali began to see the fundamental architecture of the universe itself. Time and space seemed to fold under the weight of the values being generated. The Final Calculation

Let’s see what happens:

Ordinals are not integers. The calculator must support: This script acts as a symbolic calculator

Modern development is pushing FGH calculators into new domains:

It is well known that

The calculator allows users to:

It provides a universal "yardstick." Instead of debating whether Graham's Number or TREE(3) is bigger, mathematicians determine their positions on the hierarchy.

, it has surpassed , once famous for being the largest number ever used in a serious mathematical proof. Stepping into Infinity (

[ f_\omega(2) = f_\omega[2](2) = f_2(2) = 2 \cdot 2^2 = 8 ] This program will stop if the number becomes

[ \beginaligned f_\omega+2(3) &= f_\omega+1^3(3) \ &= f_\omega+1(f_\omega+1(f_\omega+1(3))) \ f_\omega+1(3) &= f_\omega^3(3) \ f_\omega(3) &= f_3(3) \quad (\textsince \omega[3]=3) \ f_3(3) &= f_2^3(3) \dots \endaligned ]