Here I will ordinal-extend a few other odd notations.

Steinhaus-Moser notation

Steinhaus-Moser is actually just a fast iteration hierarchy with \(f_0(n) = n^n\), where n in an m-gon is \(f_{m - 3}(n)\). Diagonalizing over this we get \(f_\omega(n) = f_{n - 3}(n)\), and in general the following hierarchy:

  • \(M_3(n) = n^n\)
  • \(M_{\alpha + 1}(n) = M_\alpha^n(n)\) for \(\alpha \geq 3\)
  • \(M_\alpha(n) = M_{\alpha[n]}(n)\) for \(n \geq 3\)

I am baffled by Steinhaus' choice of a circle for \(M_5(n)\), and I suggest using \(M_\omega\) for the circle instead. And yes, we can go into \(M_{\epsilon_0}\) and \(M_{\Gamma_0}\) and whatnot.

Factorials (Torian)

  • \(n!_0 = n\)
  • \(n!_{\alpha + 1} = n!_\alpha \cdot (n - 1)!_\alpha \cdots 2!_\alpha \cdot 1!_\alpha\)
  • \(n!_\alpha = n!_{\alpha[n]}\)

So \(n!_\omega = n!_n\), which is the Torian. For \(n!_{\varepsilon_0}\) I suggest the name "Telian."

Chained Arrow

Coming soon!

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