The middle-growing hierarchy is a hierarchy created by Googology Wiki user Ikosarakt1.[1]

The rules are as following:

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

Although it is not clarified in the original definition, \(\alpha\) denotes a countable ordinal equipped with a fixed system of fundamental sequences of limit ordinals up to \(\alpha\), and \(n\) denotes a natural number.

We can see that the only difference between the middle-growing hierarchy and the fast-growing hierarchy is that \(m_{\alpha+1}(n) = m_{\alpha}^2(n)\) while \(f_{\alpha+1}(n) = f_{\alpha}^n(n)\), where \(m_{\alpha}(n)\) denotes \(m(\alpha,n)\) and the superscripts \(^2\) and \(^n\) represent an iteration of the functions \(m_{\alpha}\) and \(f_{\alpha}\).

Up to \(\omega^\omega\)

\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) &>& \textrm E[2]n\#(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.

Specific numbers

135 is equal to m(ω, 7), and also the number of nominal AM radio frequencies (n × 9 kHz), where 17 ≤ n ≤ 31 or 59 ≤ n ≤ 178) in Europe, and the bandwidth of the longwave radio band (in kHz).

264 is equal to m(ω, 8), and also approximately the number of U.S. gallons in a cubic metre, and the highest possible game value (Grand ouvert with all four jacks) in the German card game of Skat.

Sources

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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