User blog:Alemagno12/Yet another attempt at well-defining Pi function

Definition
In Pi function, instead of having the usual types of ordinals (zero, successor ordinal and limit ordinal), we have a fourth type of ordinal, called illusory ordinal. Illusory ordinals are basically the same as limit ordinals, being the limit of a sequence of ordinals (the illusory ordinal's FS, which we will denote with x{n} instead of x[n]) - however, making some ordinals illusory ordinals instead of limit ordinals is necessary for the notation to work. We also introduce a new function E(a,b) that produces illusory ordinals based on a, which we will use to simulate Pi function's transfinite FSes like ω[ω2].

Define:
 * π0(0;0) = 1
 * π0(0;1) = ω
 * πx(0;1) = Ωx
 * cof(0) = 0
 * cof(π(0;0)) = 0
 * cof(x+y) = cof(y)
 * cof(x) = 0: cof(Ωx) = Ωx
 * cof(x) > 0: cof(Ωx) = cof(x)
 * cof(y) = 0: cof(E(x,y)) = ω
 * cof(y) > 0: cof(E(x,y)) = cof(y)
 * [WIP]