Large Numbers - Systematically Creating Ever Faster Increasing Sequences

Systematically Creating Ever Faster Increasing Sequences

Given a strictly increasing integer sequence/function (n≥1) we can produce a faster growing sequence (where the superscript n denotes the nth functional power). This can be repeated any number of times by letting, each sequence growing much faster than the one before it. Then we could define, which grows much faster than any for finite k (here ω is the first infinite ordinal number, representing the limit of all finite numbers k). This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.

For example, starting with f₀(n) = n + 1:

• f₁(n) = f₀n(n) = n + n = 2n
• f₂(n) = f₁n(n) = 2nn > (2 ↑) n for n ≥ 2 (using Knuth up-arrow notation)
• f₃(n) = f₂n(n) > (2 ↑)n n ≥ 2 ↑2 n for n ≥ 2.
• f_k+1(n) > 2 ↑k n for n ≥ 2, k < ω.

• f_ω(n) = f_n(n) > 2 ↑n - 1 n > 2 ↑n − 2 (n + 3) − 3 = A(n, n) for n ≥ 2, where A is the Ackermann function (of which f_ω is a unary version).
• f_ω+1(64) > f_ω64(6) > Graham's number (= g₆₄ in the sequence defined by g₀ = 4, g_k+1 = 3 ↑g_k 3).
  • This follows by noting f_ω(n) > 2 ↑n - 1 n > 3 ↑n - 2 3 + 2, and hence f_ω(g_k + 2) > g_k+1 + 2.

• f_ω(n) > 2 ↑n - 1 n = (2 → n → n-1) = (2 → n → n-1 → 1) (using Conway chained arrow notation)
• f_ω+1(n) = f_ωn(n) > (2 → n → n-1 → 2) (because if g_k(n) = X → n → k then X → n → k+1 = g_kn(1))
• f_ω+k(n) > (2 → n → n-1 → k+1) > (n → n → k)
• f_ω2(n) = f_ω+n(n) > (n → n → n) = (n → n → n→ 1)
• f_ω2+k(n) > (n → n → n → k)
• f_ω3(n) > (n → n → n → n)
• f_ωk(n) > (n → n → ... → n → n) (Chain of k+1 n's)
• f_ω2(n) = f_ωn(n) > (n → n → ... → n → n) (Chain of n+1 n's)