User blog comment:Hyp cos/Analysis - BEAF, FGH and SGH (part 3)/@comment-2033667-20141013055709/@comment-11227630-20141013134110

What about matching ordinals and their fundamental sequences to Taranovsky's notation? Taranovsky's notation goes very far, and further than where BEAF goes, so it's strong enough.

For an ordinal $$\alpha$$ (no matter how we write it or express it) in Taranovsky's notation's range, write down its postfix form, then its length must be an odd number. Let L be a map from ordinals in Taranovsky's notation's range to integers such that the length of postfix form of $$\alpha$$ is $$2L(\alpha)+1$$.

Then define $$\alpha[n]$$ as follows: it's the greatest ordinal $$\beta$$ such that $$\beta<\alpha$$ and $$L(\beta)\leq L(\alpha)+n$$.

But there're some problems. For example, the postfix form of $$\varepsilon_0$$ can be 0Ω1C in 1st system (so $$L(\varepsilon_0)=1$$), 00Ω2CC in 2nd system (so $$L(\varepsilon_0)=2$$), 000Ω3CCC in 3rd system (so $$L(\varepsilon_0)=3$$), and so on.

To avoid this, we must use the same system all the time. We can use 1st system, 2nd system, or 100st system. In different notation systems we have different versions of fundamental sequences.