Explicit Formulas for the Jump of Q-Degrees

Author: Crawshaw, Mark

Year: 1985

Degree: Dissertation (Ph.D.)

Advisor: Kechris, Alexander S.

Committee Members: Becker, Howard Starr; Lorden, Gary A.; Luxemburg, W. A. J.; Wales, David B.; Kechris, Alexander S.

Option: Mathematics

DOI: 10.7907/b0x4-c027

Abstract

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 some forcing 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.

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