It’s the not fast-growing hierarchy!

What isn’t fast-growing?

Slow-growing hierarchy!

Yes, it actually exists.

Define g to be:

• g₀(n) = 0
• g_a+1(n) = g_a(n) + 1
• g_α(n) = g_α[n](n) where α is a limit ordinal

Now, if your first thought was “Who needs this?” then you’re not alone. After all, g_m(n) is just m, as long as m is an integer. At g_ω(n), it reaches n, and operations like ω2, ω + 1, ω^ω and similar ones are all valid, and it’s like applying the operation to n, so for g_{ω + 1}(n), it equals n + 1.

Where it gets interesting is ε₀. At g_ε0(n), it becomes n↑↑n. You can combine ε₀ and either other integers or ω to change the result. For ε_m, it becomes n↑↑((m+1)n), and for ε_ω, it is n↑↑(n^2).

Remember the next step? ζ₀? Pentation (n↑↑↑n). Next? η₀? Hexation (n↑↑↑↑n). That constant with the really big name? Γ₀? It’s about {n, n, 1, 2} in BAN (Bird’s Array Notation). ϑ(ε_Ω+1)? the largest thing I could find? X↑↑X&n. I don’t even know what it means.

How slow is SGH compared to FGH? Let’s look at values for ε₀. f_ε0(n) > {n, n[[1]]2}, whatever this means. g_ε0(n)? It’s just n↑↑n. I’ll go into detail on Bird’s Array Notation in the next post.

Until then.