User blog:Edwin Shade/A Question Concerning Ordinals (In Particular, Epsilon-Nought)

So as to improve my experience in manipulating ordinals, I decided to take an ordinal, and apply the following process to it:


 * If the ordinal is a limit ordinal then find the nth term in the fundamental sequence of that ordinal.


 * If the ordinal is a successor ordinal, subtract one.


 * Repeat until you reach 0.

This picture represents my progress in the expansion of \(\varepsilon_0\), where the 4th term of the fundamental sequence of an ordinal is to be taken when a limit ordinal is encountered. Thus far I have reduced these ordinals 118 times, and it does not look like they will terminate within a reasonable amount of time, (though I am aware of course it eventually must terminate).



For the reduction of \(\varepsilon_0\) with the 3rd element in the fundamental sequence taken each time a limit ordinal is encountered, it only takes a bit more than two score steps, yet for this it seems it might take thousands. Is there a non-recursive equation that can tell me how long \(\varepsilon_0\) will take to reduce to 0 when the n-th term of the fundamental sequence of a limit ordinal is taken, as in the above rules ? Specifically, I would like to know how many steps it would take to reduce my particular case to 0.