User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-35470197-20190912034735/@comment-31580368-20190912144236

I think we are confusing different axioms.

Ordinal is admissible if it is a transitive model of KP, then ω is 1st admissible. If we introduce the existence axiom ω, then we get KPω, and ω_1^CK is a transitive model of KPω, e.t.c. Then ω-th admissible is a transitive model of KPl. Axiomatic recognition of the existence of a certain large countable ordinal leads to an increase in it transitive model. However, the existence of a collection is a weaker axiom than the existence of the limit of this collection, but still the limit is not transitive model of theory with collection.