User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-35470197-20190912034735/@comment-35470197-20190912144817

> Ordinal is admissible if it is a transitive model of KP,

No... Ordinals are unable to be a model of KP...

> ω_1^CK is a transitive model of KPω

No... It does not satisfy the axiom of pairing...

Now according to your explanation, you are completely misunderstanding the notion of a model and the augumentation of KP set theory. Could you please check the definition of the notion of a model again? Then you will soon find your mistake. Or maybe you caught wrong description of them somewhere in this wiki.