Can anyone point me to any proofs (pref. interesting ones!) that make good (or bad) use of the Finite or Infinite Priority Injury Argument?

**Edit:** I would suppose that my question could be put this way too. Are priority injury method proofs limited to recursion theory, or have they been used elsewhere?

Motivation: It's a technique that crops up a lot in recursion/computability theory, especially in the Friedberg-Muchnik theorem. As a development of normal Priority Arguments (set up by Kleene and Post), I wish to explore any interesting, or just additional, formulations.

As I said, I'm familiar with 'Movable Marker' proofs, and with 'Priority Method' proofs, and I'm looking for proofs that make use of the injury side of the method.

For those who are unsure; the priority injury method utilises the notion that for a set of requirements that we have to meet, $R_{2e}$ for one side and $R_{2e+1}$ on the other side of our computation, we define the 'use' of each side, and then choose a *witness* $x$ s.t. $A(x)\neq \Phi^B_i (x)$, where $A$ and $B$ are the sets that we're trying to make incomparable in the Friedberg-Muchnik theorem. The key point is that we allow ourselves to finitely/infinitely *injure* the requirements that have been satisfied before so that we can satisfy a stronger requirement - it is *this* technique that I'm interested to see further examples of...

Any proofs considered!

With thanks, M.

anywriting of Post or Kleene on what you call "normal Priority arguments"; could you please specify a source? (2) there is a LARGE literature on priority arguments, with all sorts of results, part of which are included in the Soare text mentioned by Paseman in his comment above, so your question does not strike me as sufficiently focused. – Ali Enayat Jun 16 '11 at 1:24Recursively Enumerable Sets and Degrees(Perspectives in Mathematical Logic, Springer-Verlag, 1987). Note that Soare will eventually publish a new book to replace this one - people.cs.uchicago.edu/~soare/cta – François G. Dorais♦ Jun 16 '11 at 5:13