User blog comment:Syst3ms/YAUD/@comment-30754445-20180810132619/@comment-32697988-20180810161917

"Your rules explicitly talk about things like cofinality and fundamental sequences, which are clearly statements in "ordinalese"."

Does that mean if we don't use the words like "cofinality" and "fundamental sequences" and write cof(α) L(α) and write α[β] S(α,β) it will not be "ordinalese"?

You wrote "You require them to have exactly the properties of the actual cardinals I and M" but that is only true when you assume cof(α) means the cofinality of α and α[β] means the βth term in the fundamenta sequence of α. If you treat them as "some kind of function" then it is not requiring I and M to have the properties of (weakly) inaccessible cardinals and (weakly) Mahlo cardinals.