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The middle-growing hierarchy is a hierarchy created by Googology Wiki user Ikosarakt1.

The rules are as following:

  • \(m(0,n)=n+1\)
  • \(m(\alpha+1,n)=m(\alpha,m(\alpha,n))\)
  • \(m(\alpha,n)=m(\alpha[n],n)\)

Up to \(\omega^\omega\)Edit

\begin{eqnarray*}
m(0,n) &=& n + 1 \\
m(1,n) &=& n + 2 \\
m(2,n) &=& n + 4 \\
m(3,n)  &=& n + 8 \\
m(k,n) &=& n + 2^k \\
m(\omega,n) &=& n + 2^n \\
m(\omega+1,n) &=& n + 2^n + 2^{n + 2^n} \\
m(\omega+2,n) &=& n + 2^n + 2^{n + 2^n} + 2^{n + 2^n + 2^{n + 2^n}} > 2^{2^{2^n}} \\
m(\omega+m,n) &>& 2^{En\#(m+1)} > 2\uparrow\uparrow(m+1) \\
m(\omega2,n) &>& 2\uparrow\uparrow(n+1) \\
m(\omega3,n) &>& 2\uparrow\uparrow\uparrow(2^n) \\
m(\omega m,n) &>&  2\uparrow^m(2^n) \\
m(\omega^2,n)  &>& 2\uparrow^n(2^n) \\
m(\omega^2+\omega,n)  &>& \lbrace n,2^n,1,2 \rbrace \\
m(\omega^22,n)  &>& \lbrace n,2^n,n,2 \rbrace \\
m(\omega^3,n) &>&  \lbrace n,2^n,n,n \rbrace \\
m(\omega^m,n) &>& \lbrace n,m+1 (1) 2 \rbrace \\
m(\omega^{\omega},n) &>& \lbrace n,n+1 (1) 2 \rbrace > \lbrace n,n (1) 2 \rbrace \\
\end{eqnarray*}

We see that the middle-growing hierarchy catches the fast-growing hierarchy at \(\omega^{\omega}\), and generally, it does so at all multiples of it.

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