User blog comment:P進大好きbot/Rathjen-type Ordinal Notation/@comment-11227630-20190823085852/@comment-35470197-20190823092656

> So a recursive ordinal notation derived from an OCF looks much more complicated than the OCF.

It is true. We can use set theory in the definition of an OCF, which is uncomputable by definition, while we can only use arithmetic in the definition of an associated ordinal notation. Since arithmetic is much less powerful than set theory, we need such a complicated construction.