User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-31580368-20190913014700

So, here is my humble attempt to redefine the axioms of large countable ordinals for their use in the KP.

For admissible ordinal: KP+x means: KP with the axiom schema for all φ(pn)-formulas such that ∀pn(φ(pn)→∃γ(γ⊧φ(pn))), where all φ(pn)-formulas are defined as formulas that include a formula such that φ|Lx⊧φ.

For collection of admissible ordinal: KP+x means: KP with the axiom schema for all φ(pn)-formulas such that ∀pn(φ(pn)→∃γ(γ⊧φ(pn))), where all φ(pn)-formulas are defined as formulas that include collection of formula such that for every φk|Lα k ⊧φk. And x is collection of αk ordinals.

For limit of admissible ordinal: KP+x means: KP with the axiom schema for all φ(pn)-formulas such that ∀pn(φ(pn)→∃γ(γ⊧φ(pn))), where all φ(pn)-formulas are defined as formulas that include collection of formula such that for every φk|Lα k ⊧φk. And exist x=sup(αk) such every φk(pn) have limit ≤ Lx.