I am trying to prove another statement, which is the following: There is a complete extension of PA whose degree is below 0'. Moreover, one can show that there is a complete extension T of PA in each level of the hierarchy, i.e. T is turing reducible to 0^{n+1} but not to 0^{n}.
I remember a construction that I have seen before to show that given a recursive tree A then there is a branch of this tree whose degree was below 0'. The key point about this construction was to use the limit lemma and approximate this branch by stages. (changing the opinion at some point). So I guess that the same proof should work to prove the statement above. Fixing an enumeration of all the possible statements, at each stage s I can decide if I want to add the sentence B_{s} or it negation. The problem is that I don't believe that the process to check if the given sentence is consistent with the previous stages is consistent or not. (that's where 0' just come to the game, something like asking "at some point we prove something false"?). Any idea to how to formalize the argument (or actually fixed it) would be appreciated.