User blog:Ubersketch/An outline for a system.

https://googology.wikia.org/wiki/User_blog:Ubersketch/Hypernomial_representation_of_ordinals

https://googology.wikia.org/wiki/User_blog:Ubersketch/Hypernomial_Hierarchy_v2

See the above posts first.

The system isn't based on ordinals, but rather hypernomials.

The system should reach the Church-Kleene ordinal, meaning some functions in the system don't terminate, and their termination is undecidable, but some functions may be terminating such as tetration.

The system should apply pure recursion on successorship and the zero function, i.e. only repeating successorship.