User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-35470197-20190912034735/@comment-35470197-20190912140924

Then "axiom of existence admissible" is a typo of "axiom of existence of admissible". Anyway, the point is that you seem to be using KPx for the case where x is not such existence of an ordinal or a collection of ordinals. Usually, KPω^r is not KP augumented by the existence of ω^r, KPI is not KP augumented by the existence of I, and KPi is not KP augumented by such existence axiom. Is there any relation between your KPx and the usual KPx? Or am I incorrectly understanding what you wrote?