This question comes from the [Wikipedia article on Kleene's O](http://en.wikipedia.org/wiki/Kleene%27s_O#Properties_of_Paths_in) and a [previous Math Overflow question](https://mathoverflow.net/questions/71584/hyperarithmetic-statements-decidable-by-induction-up-to-a-recursive-ordinal).
The claim in Wikipedia that I have a question about is the second sentence in the following quote.
"There exist $\aleph\_0$ paths through $\mathcal{O}$ which are $\Pi^1\_1$. Given a progression of recursively enumerable theories based on iterating Uniform Reflection, each such path is incomplete with respect to the set of true $\Pi^0\_1$ sentences."
I do not understand the informal proof in the second sentence I would appreciate a more complete explanation and/or a reference.



Paths in Kleene's O and deciding $\Pi^0_1$ sentences 
The basic reference for this is Feferman and Spector, *Incompleteness Along Paths in Progressions of Theories* [JSL 27 (1962), 383-390]. Theorem 2.5 states:



> 
> If $Z$ is a path through $O$ and $Z \in \Pi$ then $Tr\_1 \nsubseteq \bigcup\_{d \in Z} S\_d$.
> 
> 
> 


Here, $\Pi$ basically means $\Pi^1\_1$ in modern notation, $Tr\_1$ is the set of true $\Pi^0\_1$ sentences, and $\{S\_d : d \in I\}$ is any progression of theories such that:


1. $O \subseteq I \subseteq \omega$;
2. If $d \in O$, then $S\_d$ is consistent;
3. If $c, d \in O$ and $c \leq\_O d$, then $S\_c \subseteq S\_d$; and
4. the relation $Thm[\psi,d]$ which holds if and only if $d \in I$ and $\psi \in S\_d$ is recursively enumerable.


Then, Theorem 3.7 states:



> 
> There exists a path $Z$ through $O$ with $Z \in \Pi$. In fact, for any $d \in O^\ast - O$, $Z = O \cap C'(d)$ is such a path.
> 
> 
> 


Here, $O^\ast$ is an extension of $O$ with some nonstandard notations, and $C'(d)$ is the set of predecessor of such a notation. (Basically, $O^\ast$ has the same definition as $O$, but one quantifies only over the hyperarithmetic subsets of $\omega$ instead of all subsets of $\omega$. Thus, elements of $O^\ast$ describe *pseudowellorderings*: linear orders that have no hyperarithmetic descending sequences. By a well-known result of Kleene, it follows that $O^\ast$ is $\Sigma^1\_1$ and therefore $O^\ast - O$ is nonempty.)


