User blog:Edwin Shade/Hyper Operations On Ordinals Using The M.O.S.T. Function

In this post I will be detailing a method of extending the hyper-operators to ordinals, which follows a very Veblen-esque route of construction. Unlike that which has been attempting thus far however, I will be extending this system to uncountable ordinals as well.

The M.O.S.T. function stands for the "Multiarrowed Ordinal Snowballing Train", which is an apt description of the nature of the function of which I describe below.


 * \(\alpha\uparrow^{n}\beta=\alpha\uparrow^{n-1}(\alpha\uparrow^{n}(\beta-1)+1)\) when \(\beta\) is the successor of a limit ordinal, this is done so as to prevent catching on the fixed points that limit normal ordinal tetration at \(\epsilon_0\) and so forth.


 * \(\alpha\uparrow^{n}\beta=\alpha\uparrow^{n-1}(\alpha\uparrow^{n}(\beta-1))\) when \(\beta\) is a successor ordinal.


 * \(\alpha\uparrow^{n}\beta=\text{sup}\{\alpha\uparrow^{n}\gamma|\gamma<\beta\}\)