I tried to define Hardy Hierarchy up to , and I got the ruleset with 31 rules. It ignores the facts like for not making technical complexities, but it doesn't matter because ordinals can be treated just as notational symbols. Also, exponentiation isn't allowed: we must define it through . So, the ruleset for which I was working for a few weeks:
These rules were written so that first rules handle smaller structures and extensions, and last otherwise.
Feel free to improve it (particularly in rule-conditions, I think they must be more formal.)
This ruleset is incorrect. The correct version is under construction.