About the recent MO question of yours, what if you just set up the system of equations relating the various \(\varphi_i\) and their derivatives and just ask for solution sets and solvability conditions via differential-geometric means? Maybe this way you could attract experts in the field of solving things, without them being problem the equation originates from.
In order to do this you could make it look as if it was a general textbook exercise maybe, totally self contained and unrelated to hyper/Goodstein and things like that.
I tried to follow the equations and your post and come up with the system myself, independently, but it felt like trimming my brain cells (I have less neurons now xD). At the intuitive level, It seems to me that there are too many nested layers of functional dependency to extract implicit functions without A) becomeing insane B) doing something wrong C) hitting some serious obstruction to the existence of a solution.
In order to do this you could make it look as if it was a general textbook exercise maybe, totally self contained and unrelated to hyper/Goodstein and things like that.
I tried to follow the equations and your post and come up with the system myself, independently, but it felt like trimming my brain cells (I have less neurons now xD). At the intuitive level, It seems to me that there are too many nested layers of functional dependency to extract implicit functions without A) becomeing insane B) doing something wrong C) hitting some serious obstruction to the existence of a solution.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)