User blog comment:Vel!/Ordinals and arrow notation/@comment-3427444-20131212014842

On Sbiis Saibian's "The Fast Growing Hierarchy below Gamma-Naught" page, it says that, with base 3, ω^(ε_0+1) takes 3^(3^^3+1) nestings to get to the base case, ω^(ω^(ε_0+1)) takes 3^3^(3^^3+1), ε_1 takes 3^3^3^(3^^3+1) or about 3^^6, ε_2 takes about 3^^9, etc. As you can see, you can change the ω's to 3's on the initial ordinal to get the number of nestings. This is similar to Bowers' & (array of) operator where the number of entries in the resulting array can be found by solving the left argument of the & operator.

This result supports Chris Bird's (and subsequently Wythagoras') theory that ε_1 could be interpreted as ω^^(ω*2) because ε_1 expands to ω^ω^ω^ ... ^ω^(ε_0+1).

In conclusion, Wythagoras's definition looks more "natural" to me.