User blog comment:MilkyWay90/My ordinal function/@comment-35470197-20180630144403/@comment-35470197-20180702131519

> I think you got a different version of it again, so let me explain in this reply.

I think that I did not got different version. (I just used the symbols \(x_i\) in order to avoid the conflict of the index symbol \(a\) and the function symbol \(a\).)

Anyway, in order to go beyond the addition, it is good to construct an ordinal notation so that you can input an ordinal number to the first variable of \(F\). For example, if you use the ordinal notation given as \begin{eqnarray*} \mathbb{N}^2 & \to & \omega^2 \\ (x_1,x_2) & \mapsto & \omega \times x_2 + x_1, \end{eqnarray*} then you obtain the following stronger definition: \begin{eqnarray*} F(0,0,a(n)) & = & a(n) \\ F(0,x_2+1,a(n)) & = & a^{F(n,x_2,a(n))}(n) F(x_1+1,x_2,a(n)) & = & a^{F(x_1,x_2,a(n))}(n) \\ \end{eqnarray*}