Fast Growing Hierarchy Calculator

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.

If you try to compute ( f_ω+1(4) ) on a standard calculator, it will crash, overflow, or freeze. Why?

Computational Explosion. Consider the fast-growing hierarchy for ( f_ω(n) ): fast growing hierarchy calculator

Now, ( f_ω+1(3) ) requires applying ( f_ω ) three times. That is ( f_ω(f_ω(f_ω(3))) ). The second iteration is already ( f_ω(7.6 \times 10^12) ). To reduce that, the computer would need to iterate ( f_7.6 \times 10^12 ) on itself. The number of steps exceeds the number of atoms in the universe.

Thus, an FGH calculator does not "finish" computing. It approximates. It translates the FGH expression into a known large number notation (Conway chained arrows, BEAF, or TREE sequence comparisons). Logicians use ordinal analysis to measure the strength

For a given f_α(n):

Let’s see what happens:

By the time we reach ( f_\omega(n) ) (where ( \omega ) is the first infinite ordinal), we’ve surpassed primitive recursive functions. By ( f_\omega+1(n) ), we’re in the realm of the Ackermann function. And then it gets fast. Now, ( f_ω+1(3) ) requires applying ( f_ω ) three times

To give you a sense:
( f_\omega^\omega(3) ) is a number so large that writing it down in standard notation would require more digits than there are particles in the observable universe—by an absurd margin.

Using an FGH calculator requires mathematical humility.