User blog:Alemagno12/Fundamental sequences for DON

Dropper Ordinal Notation was intended to be a very strong notation, but it is ill-defined. However, in this blog post, I will make fundamental sequences for DON, and I will also make it stronger.

The definitions of diagonalizers, eventually overgrows, @x and $x can be found here.
 * Diagonalizers are the same as 0-diagonalizers.
 * x+1 is a sequence ordinal.
 * @y (where y is a limit ordinal) is a limit ordinal.
 * D[x] is an omega-limit ordinal.
 * Ψ(x) = Ψε 0(x)
 * Ψx(0) = x
 * Ψx(y+1) = ΨΨ x(y) (0)
 * Ψx(y) (y is a limit ordinal) = sup(Ψx(y[1]),Ψx(y[2]),Ψx(y[3]),...)
 * D[x+1] is the 0-diagonalizer of function F(y) = ΨD[x](y)
 * D[@Dx] is the x-diagonalizer of function F(y) = D[@y]
 * If x is the y+1-diagonalizer of function F, for any function G that eventually overgrows F, G(@x) is the y-diagonalizer of function H(y) = G(@F(y))