For finite ordinals (normal whole numbers), the next function is defined as the iteration of the previous one. $$f_k+1(n) = f_k^n(n)$$ Note: The superscript denotes iteration, not exponentiation. $f_k^n$ means applying the function $f_k$ to $n$ a total of $n$ times.
f_ω*4(3) → f_ω*3 + ω(3) → Systems expands to f_ω*3 + 3(3) → eventually becomes f_ω*3 + 2(f_ω*3+2(f_ω*3+2(3))) fast growing hierarchy calculator
$$f_2(n) = f_1^n(n)$$ This iterates doubling. $f_2(n)$ roughly equates to multiplication, leading to $n \cdot 2^n$. In the context of standard hierarchy calculators, this often corresponds to exponential growth. For finite ordinals (normal whole numbers), the next
Users often want to see how these functions relate to famous named numbers. Benchmark Mapping: Automatically flag when a calculation surpasses Graham’s Number Ackermann function Notational Translation: Convert results into Knuth’s Up-Arrow Notation Conway Chained Arrow Notation Bowers' Exploding Array Notation where possible. 4. Custom Fundamental Sequences f_ω*4(3) → f_ω*3 + ω(3) → Systems expands
fα+1(n)=fαn(n)f sub alpha plus 1 end-sub of n equals f sub alpha to the n-th power of n This means applying the previous level's function fαf sub alpha to the input : For a limit ordinal
4.5/5 Best for: Googology enthusiasts, math students exploring computability, and anyone curious about how absurdly large numbers are defined.
Look for online implementations like "Googology FGH Simulator" or "Ordinal Calculator by Deedlit." Input f_ε₀(3) and watch as infinite recursion folds into a finite, humbling, display of 3↑↑... . That is the beauty of the fast-growing hierarchy: it lets us touch the infinite, one recursion at a time.