# Hardy Hardy Hardy Wainer Wainer Wainer

Hierarchy Hierarchy Hierarchy

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:

• H0(n) = n
• Ha+1(n) = Ha(n+1)
• Ha(n) = Ha[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:

• H0(n) = n
• H1(n) = n + 1
• H2(n) = n + 2
• Hω(n) = 2n
• Hω + 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 Ha+1(n) = Ha(n+1). So for a = ω, Hω + 1(n) = 2(n + 1) = 2n + 2.

Continuing!

• Hω + m(n) = 2(n + m)
• Hω2(n) = 4n
• Hω3(n) = 8n
• Hωm(n) = 2mn
• Hω2(n) = 2nn (remember? this is equivalent to f2(n))
• Hω3(n) ≈ n↑↑n

You know what? I’ll link the website.

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
• ωa+1[n] = ωan where ωan = ωa + ωa + ωa + ωa + ωa + ωa + ωa + ωa + ωa + … n times
• ωa[n] = ωa[n] if and only if a is a limit ordinal
• a1a2a3a4a5+…+ωak)[n] = ωa1a2a3a4a5+…+ωak[n] where a1≥a2≥…≥ak
• ε0 = 0 (alternatively 1)
• ε0[n+1] = ωε0[n] = ω ↑↑ (n-1) (alternatively ω ↑↑ n)

It’s oddly weird.

Next time, I’ll talk about the Veblen and Buchholz hierarchies.

Toodle-oo!

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