User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-35470197-20190914133340

Although Θ will be well-defined for recursive ordinals, this formulation does not satisfy them. In particular, PTO(PA+Ind(ω)) = ε_0, and Ind(ω_1^CK) is ill-defined. (Please remember that admissible ordinals are recursive analogues of cardinals, but are not recursive.)