Hardy hierarchy. You remember fast- and slow-growing hierarchy? It’s another one of those.

Honestly, it feels as slow as SGH.

Hardy hierarchy is defined as follows:

H_{0}(n) = n

H_{a+1}(n) = H_{a}(n+1)

H_{a}(n) = H_{a[n]}(n) if a is a limit ordinal

I haven’t fully explained what limit ordinals are, so in the future posts, I’ll explain.

Evidently:

H_{0}(n) = n

H_{1}(n) = n + 1

H_{2}(n) = n + 2

…

H_{ω}(n) = 2n

H_{ω + 1}(n) = 2n + 2

Before we continue, why isn’t it 2n + 1, like other hierarchies? Because the second rule of Hardy hierarchy states that H_{a+1}(n) = H_{a}(n+1). So for a = ω, H_{ω + 1}(n) = 2(n + 1) = 2n + 2.

Continuing!

H_{ω + m}(n) = 2(n + m)

H_{ω2}(n) = 4n

H_{ω3}(n) = 8n

H_{ωm}(n) = 2^{m}n

H_{ω2}(n) = 2^{n}n (remember? this is equivalent to f_{2}(n))

Wainer hierarchy is actually in the FGH website, so I’ll link the section. Fun fact: when people say “Fast Growing Hierarchy”, they can refer to different hierarchies. I’ll cover other hierarchies in future posts.

Wainer Hierarchy is defined as the following:

ω[n] = n

ω^{a+1}[n] = ω^{a}n where ω^{a}n = ω^{a} + ω^{a} + ω^{a} + ω^{a} + ω^{a} + ω^{a} + ω^{a} + ω^{a} + ω^{a} + … n times

ω^{a}[n] = ω^{a[n]} if and only if a is a limit ordinal

(ω^{a1}+ω^{a2}+ω^{a3}+ω^{a4}+ω^{a5}+…+ω^{ak})[n] = ω^{a1}+ω^{a2}+ω^{a3}+ω^{a4}+ω^{a5}+…+ω^{ak}[n] where a_{1}≥a_{2}≥…≥a_{k}