User blog:Kyodaisuu/Approximation of array notation with FGH and Hardy hierarchy

In the page of FGH, array notation is approximated as

\begin{eqnarray*} f_{\omega^2}(n) &>& \lbrace n,n,n,n \rbrace \\ f_{\omega^3}(n) &>& \lbrace n,n,n,n,n \rbrace \\ \end{eqnarray*}

I wanted to get approximation which describes all ordinals below \(\omega^\omega\), such as the equation in Taro's multivariable Ackermann function, as follows.

\[A(..., a3, a2, a1, a0, n) & \approx & f_{... + \omega^3･a3 + \omega^2･a2 + \omega･a1 + a0}(n)\]

By discussing with Aetonal, I noticed the relationship that

\(\lbrace n,n,n,n,n \rbrace = \lbrace n,2,1,1,1,2 \rbrace \)

and therefore

\begin{eqnarray*} f_{\omega^2}(n) &>& \lbrace n,n,n,n \rbrace = \lbrace n,2,1,1,2 \rbrace \approx A(1,0,0,n) \\ f_{\omega^3}(n) &>& \lbrace n,n,n,n,n \rbrace = \lbrace n,2,1,1,1,2 \rbrace \approx A(1,0,0,0,n) \\ f_{... + \omega^3 a_3 + \omega^2 a_2 + \omega a_1 + a_0}(n) &>& \lbrace n,2,a_0+1,a_1+1,a_2+1,a_3+1,... \rbrace \approx A(..., a_3, a_2, a_1, a_0, n) \\ \end{eqnarray*}

After that, I wondered about the second entry of the array notation, because the above approximation only shows the case where the 2nd entry is 2. By further discussion, I realized that the array notation can be better approximated with Hardy hierarchy as follows.

\[\lbrace n,a_0+1,a_1+1,a_2+1,a_3+1,... \rbrace \approx H_{\omega^{... + \omega^2 a_3 + \omega a_2 + a_1} a_0}(n)\]