Fast Growing Hierarchy Calculator |verified| 〈PREMIUM〉

Fast-Growing Hierarchy

To create a calculator for the (FGH), you must implement a recursive system based on an ordinal-indexed family of functions

# Base Case: f_0(n) = n + 1 if alpha == 0: return n + 1 fast growing hierarchy calculator

• f0(n) = n+1
• f1(n) = f0 iterated n times on n = n + n = 2n (verify small n: f1(1)=f0(1)=2, f1(2)=f0(f0(2))=4 -> equals 2n)
• f2(n) = f1 iterated n times on n. Since f1(x)=2x, f2(n)=2^n * n (i.e., multiply by 2 each iteration) — roughly exponential times n.
• f3(n) = iterate f2 n times on n → roughly a tower of exponentials of height n (Ackermannian growth).
• fω(n) = f_n(n) whose growth surpasses any fixed finite level; already for moderate n values this is astronomically large.

9. Deliverables (if you want me to implement)

| Function | Formula | Calculator Input | Result | | --- | --- | --- | --- | | F1 | n + 1 | n = 5 | 6 | | F2 | 2n | n = 5 | 10 | | F3 | 2^n | n = 5 | 32 | | F4 | 2^(2^n) | n = 5 | 2^(2^5) = 2^32 = 4,294,967,296 | Fast-Growing Hierarchy To create a calculator for the

• f0(n) = n + 1
• fα+1(n) = fα iterated n times on n (i.e., fα^(n)(n))
• fλ(n) = fλn for limit λ where λ[n] is a fundamental sequence approaching λ

• f_0(2) = 3
• f_1(2) = f_0^(2)(2) = 4
• f_2(2) = f_1^(2)(2) = f_1(f_1(2)) = f_1(4) = 5 → 5 (small)
• f_3(2) ≈ iterated application producing modest numbers
• f_ω(2) equals Ackermann(2,2)-scale — present as "≈ A(2,2) scale" or "tetration-level" (Note: specific higher examples are symbolic due to extreme growth.)