In the context of the axiom of projective determinacy, Q-degrees have been proposed as the appropriate generalisations of the hyperdegrees to all the odd levels of the projective hierarchy. In chapter one we briefly review the basics of Q-theory.
In the second chapter we characterise the Q-jump operation in terms of certain two-person games and derive an explicit formula for the Q-jump. This makes clear the similarities between the Q-degrees and the constructibility degrees, the Q-jump operation being a natural generalisation of the sharp operation.
In chapter three we mix our earlier results with someforcing techniques to get a new proof of the jump inversion theorem for Q-degrees. We also extend some results about minimal covers in hyperdegrees to the Q-degrees. Many of our methods are immediately applicable to the constructible degrees and provide new proofs of old results.
