User blog comment:Nayuta Ito/The attempt to make my own ordinals collapse function/@comment-28606698-20170423081909

1) You did not describe in definition how to work in case if argument is uncountable. Output of your function is defined as smallest number greater than all possible results of operations of addition and multiplication with all outputs of your function with an arguments less than the current. But all countable ordinals (for example, $$\varepsilon_0$$, $$\varepsilon_0+1$$, $$\zeta_0$$, $$\Gamma_0$$, SVO, LVO and so on) are less than $$\Omega$$ since it is first uncountable ordinal. To avoid such kinf of  undetermination  one usually writes in definitions of OCF that arguments of functions in  set $$C_n$$ also are belong to previous set.

2) if you will describe function such that $$\rho(\Omega)=\varepsilon_0$$ and $$\rho(\Omega 2)=\varepsilon_1$$  then $$SVO=\rho(\Omega^{\Omega^\omega})$$ and $$LVO=\rho(\Omega^{\Omega^\Omega})$$