User blog comment:Alemagno12/I made a Pi w-reflection OCF, how horribly wrong am I?/@comment-35470197-20180812220655/@comment-35470197-20180813143402

> Oh. Well, we can define a notation system \(T(\Pi^2_0)\) for the OCF as follows:

Yeah, you can define a notation system. But it is not an ordinal notation system. Do you have a recursive well-order on the system?

> Why not?

Let \(T\) denote your notation system for short. Then you have a natural map \(o \colon T \to \textrm{Ord}\). Put \(X := \sup (\textrm{Im}(o) \cap \Omega)\). Then you have \(X \in \Omega\) by the countability of \(T\).

If you expects \(\alpha = \sup_{n < \omega} \alpha[n]\) for any limit ordinals below \(X\), you need to show that \(X = \textrm{Im}(o) \cap \Omega\). Namely, there is no "skip" in the collection of countable ordinals presentable by your functions. This problem is what I explained here more in detail.

https://googology.wikia.com/wiki/User_blog:P%E9%80%B2%E5%A4%A7%E5%A5%BD%E3%81%8Dbot/Relation_between_an_OCF_and_an_Ordinal_Notation#Strength