This question comes from the <a href="http://en.wikipedia.org/wiki/Kleene%27s_O#Properties_of_Paths_in">
Wikipedia article on Kleene's O</a> and a <a href="https://mathoverflow.net/questions/71584/hyperarithmetic-statements-decidable-by-induction-up-to-a-recursive-ordinal">previous Math Overflow question</a>.
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.