$ \def \PRo {{\mathcal { PR } _ \omega}} $ The class of primitive recursive ordinal functions in the constant omega function (henceforth denoted by $ \PRo $) are defined by Jensen and Karp (1971) as the smallest class of functions mapping finite tuples of ordinals to ordinals, which contains the constant omega function, the constant zero function, the successor function, the order testing function and all projections, and is closed under substitution and primitive recursion. More precisely, we have the following.

All the initial functions of the following forms are in $ \PRo $:

- $ \Omega ( \alpha ) = \omega $
- $ Z ( \alpha ) = 0 $
- $ S ( \alpha ) = \alpha + 1 $
- $ C ( \alpha , \beta , \gamma , \delta ) = \begin {cases} \alpha & \gamma < \delta \\ \beta & \gamma \ge \delta \end {cases} $
- $ P ^ n _ m ( \alpha _ 1 , \dots , \alpha _ n ) = \alpha _ m $, for all positive integers $ n $ and $ m $ with $ m \le n $
Every function defined by means of the following rules using previously constructed functions in $ \PRo $ is itself in $ \PRo $:

- $ f ( \alpha _ 1 , \dots , \alpha _ m , \beta _ 1 , \dots , \beta _ n ) = g \big( h ( \alpha _ 1 , \dots , \alpha _ m ) , \beta _ 1 , \dots , \beta _ n \big) $
- $ f ( \alpha _ 1 , \dots , \alpha _ m , \beta _ 1 , \dots , \beta _ n ) = g \big( \alpha _ 1 , \dots , \alpha _ m , h ( \alpha _ 1 , \dots , \alpha _ m ) , \beta _ 1 , \dots , \beta _ n \big) $
- $ f ( \alpha _ 1 , \dots , \alpha _ m , \beta ) = g \big( \alpha _ 1 , \dots , \alpha _ m , \sup _ { \gamma < \beta } f ( \alpha _ 1 , \dots , \alpha _ m , \gamma ) \big) $

Let's say an ordinal number $ \alpha $ is *$ \PRo $-reachable* whenever there is a unary function $ f $ in $ \PRo $ such that $ f ( 0 ) = \alpha $.

My question is how to characterize the $ \PRo $-reachable ordinal numbers. What is the supremum of the $ \PRo $-reachable ordinals? What is the least ordinal not $ \PRo $-reachable? Do they coincide (i.e. is the class of $ \PRo $-reachable ordinals downward closed)?

I have only the simple observation that each $ \PRo $-reachable ordinal must be countable, and as there are only countably many functions in $ \PRo $, not all countable ordinals are $ \PRo $-reachable. I also suspect that there is a connection with $ \omega _ 1 ^ { \text { CK } } $, as $ \PRo $ is recursively defined, but this feeling is a loose one.

*Jensen, Ronald B.; Karp, Carol*, Primitive recursive set functions, Axiomatic Set Theory, Proc. Sympos. Pure Math. 13, Part I, 143-176 (1971). ZBL0221.02027.