## FANDOM

10,818 Pages

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."

Coming soon!