User blog comment:B1mb0w/Fundamental Sequences/@comment-5529393-20160124015439/@comment-10262436-20160206071137

My motivation to add Rule 2B was to avoid ambiguity. Lets review Deedlit11's example

phi(2,0)^^w = e_{phi(2,0)+1}

This is either ambiguous or adds no explaanatory power. If we apply the rule e_n = e_{n-1}^^w then we get:

phi(2,0)^^w = e_{phi(2,0)+1} = e_{phi(2,0))^^w

in other words;  phi(2,0) = e_{phi(2,0)}  which appears to be correct and obvious.

The same result is reached if we apply: phi(2,0) = phi^w(1,0_*)

phi(1, phi(2,0)) = phi(1,phi^w(1,0_*)) = phi^{1+w}(1,0_*) = phi^w(1,0_*) = phi(2,0)

Have I made a mistake here ? The logic seems sound.

Appreciate your thoughts on this.