Fast — Growing Hierarchy Calculator
Beyond Infinity: The Quest for a Fast-Growing Hierarchy Calculator
Introduction: The Number That Broke the Universe
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.
Example Calculation: ( f_ω+1(3) )
- Step 0: ( f_ω+1(3) )
- Definition: ( f_ω+1(3) = f_ω(f_ω(f_ω(3))) )
- Inner: ( f_ω(3) = f_3(3) ) (Already a massive number: 3↑↑↑3 = 7.6 trillion digits)
- Result: The calculator will output: ( f_3(f_3(3)) ) or "~ 3 ↑↑↑ (3 ↑↑↑ 3)"
- Base Level (f_0): ( f_0(n) = n + 1 ) (Increment)
- Successor Level (f_α+1): ( f_α+1(n) = f^n_α(n) ) (Iteration)
- Limit Level (f_λ): For a limit ordinal λ, ( f_λ(n) = f_λ[n](n) )
Fast-Growing Hierarchy (FGH) is an ordinal-indexed family of rapidly increasing functions, fast growing hierarchy calculator
