User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-35470197-20190912034735/@comment-35470197-20190912142131

> If we are talking about a specific ordinal, then you can think of these ordinals not as additions, but as axiom "the universe is this ordinal".

Really...? Then in your context, right?
 * KPω is KP augumented by "the universe is ω",
 * KPω^r is KP augumented by "the universe is ω^r", and
 * KPI is KP augumented by "the universe is I",