User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-35470197-20190913043640

I guess that you are using wrong grammer of mathematical English. Phrases like "{symbol} {noun}" in your description should mean "{noun} {symbol}". For example, you need to use "ordinal α" instead of "α ordinal".

> for all φ(p_n)-formulas

What does "φ(p_n)-formula" mean? Is it a hierarchy such as "Π_n-formula" or a declaration of the formal language? Since you quantify p_n in the description of the axiom schema, the occurrence of p_n here is strange. Also, could you precisely quantify p_n (and also γ)? Is it "for all natural numebr p_n" (then what is n?), "for all sequence (p_n)_n", "for all set p_n", or something like that? The unbounded quantification means "for all set", but I guess that it is not what you intend. Do you intend "formula φ(p_n) on a sequence (p_n) of natural numbers"?

> where all φ(p_n)-formulas are defined as

The description is not even a definition. I guess that your use of "all" is not the correct one in mathematical English. Do you intend the following?


 * For admissible ordinal: KP+x means:
 * KP augumented by the following instance of the axiom schema for every formula \(\phi(p_n)\) on a sequence \((p_n)\) of natural numbers such that \(L_x \models \phi(p_n)):
 * \(\phi(p_n) \to \exists \gamma(\gamma \models \phi(p_n)\)

This description still does not make sense, because φ(p_n) should be closed and hence there should not be a free occurrence of \(p_n\), and the existence of \(\gamma\) immediately follows from the assumption \(L_x \models \phi(p_n)\).

In order to avoid the ambiguity, please use precise mathematical English, and precise quantification of n, p_n, γ, and k.