User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-35470197-20190912034735/@comment-31580368-20190912135557

1) I accept all the axioms of KP (without the axioms of existence of ordinals), and in addition to them we can add the axiom of the existence of some large countable ordinal from the list. All these axioms (except # 2 - ∃ω) have the property of transitivity. If I accept axiom # 7, then axioms # 3-6 are redundant, they follow from axiom # 7.

In addition, the list contains not only axioms about the existence of a particular ordinal, but also axioms about the existence of a collection of ordinals. For example, axiom "∀n∈ω∃n-th admissible" is weaker than axiom "∃limit of (n<ω)-th admissible", but stronger than any axiom "∃n-th admissible".

The use of the term "n∈" indicates precisely the axioms of the existence of a collection of ordinals.

2) I agree that such a definition is incorrect, I fixed it on "eventual domination".