This question comes from the Wikipedia article on Kleene's O and a previous Math Overflow question. 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.