User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-31580368-20190913103555

Sorry for my mathematical English. p_n - means veсtor in these definitions. But anyway indeed, these definitions are redundant and not correct. I agree that Lx⊧φ → ∃γ. And this is a tautology.

Then maybe it’s worth giving completely simplified definitions without parameters?

For admissible ordinal: KP+x means: KP augumented by the following instance of the axiom schema for every formula φ such that φ|Lx⊧φ.

For collection of admissible ordinal ("n∈O"-term): KP+x means: KP augumented by the following instance of the axiom schema for every formula φ such that exist φ|Ln⊧φ for every admissible n < O, where x is collection of admissible ordinals < O.

For limit of admissible ordinall ("n<O"-term): KP+x means: KP augumented by the following instance of the axiom schema for every formula φ such that exist φ|Ln⊧φ for every admissible n < O. And exist x=sup(n) such every φ have limit ≤ Lx.