FANDOM


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.

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.