Explicit Formulas for the Jump of Q-degrees

• Main Author: Crawshaw, Mark
• Format: Others
• Language: en
• Published: 1985
• Online Access: https://thesis.library.caltech.edu/10020/1/Crawshaw_M_1985.pdf; Crawshaw, Mark (1985) Explicit Formulas for the Jump of Q-degrees. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/b0x4-c027. https://resolver.caltech.edu/CaltechTHESIS:01232017-115539334 <https://resolver.caltech.edu/CaltechTHESIS:01232017-115539334>

<p>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.</p> <p>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.</p> <p>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.</p>