User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-35470197-20190913121749

The simplified one still does not work. It might be confusing, but in the declaration of axioms, we are not allowed to use the resulting theory itself. For example, you tried to declare the axiom schema using the condition \(L_x \models \phi\). This condition is not decided by its syntax, because it is a statement in the set theory itself. You can only use syntax theoretic conditions (e.g. "\(\phi\) is of length < 1000", "\(\phi\) is closed", "\(\phi\) admits at most one free variable", and so on) in order to declare axioms. In order to fix the issue, you need to consider the instance "\((L_x \models \phi) \to \phi\)" for any closed formula \(\phi\). But this is not what you want, because this does not yield a hierarchy of set theories such that larger ordinals correspond to stronger theories.

I am still not understanding what "n∈O"-term precisely means. What property is "n∈O"?

> for every formula φ such that exist φ|Ln⊧φ for every admissible n < O,

I could not understand this grammer. What is the intended subject of the verb "exist"?