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
  • ga+1(n) = ga(n) + 1
  • gα(n) = gα[n](n) where α is a limit ordinal

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

Where it gets interesting is ε0. At gε0(n), it becomes n↑↑n. You can combine ε0 and either other integers or ω to change the result. For εm, it becomes n↑↑((m+1)n), and for εω, 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(εΩ+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. fε0(n) > {n, n[[1]]2}, whatever this means. gε0(n)? It’s just n↑↑n. I’ll go into detail on Bird’s Array Notation in the next post.

Until then.

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