## FANDOM

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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.

$A(..., a_3, a_2, a_1, a_0, n) \approx f_{... + \omega^3･a_3 + \omega^2･a_2 + \omega･a_1 + a_0}(n)$

By discussing with Aetonal, I noticed the relationship that

$\lbrace n,2,1,1,1,2 \rbrace = \lbrace n,n,n,n,n \rbrace$

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.

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

Aetonal realized that similar expression is also possible with FGH, as follows.

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

Based on this analysis, "Approximations in other notations" was added to array notation and Taro's multivariable Ackermann function.