Cornell Math - MATH 781, Fall 2006

MATH 781: Seminar in Logic (Fall 2006)

Instructor: Richard Shore

Meeting Time & Room

As usual the seminar will focus on a special topic one day a week. Participants will take turns lecturing. The other day will be devoted to a variety of topics depending on the interests and research activities of the participants and will include some outside speakers as well.

The topic for this semester will be hyperarithmetic sets and degrees. We will continue to use the book Higher Recursion Theory by Gerald Sacks. The subject involves generalizing recursion theory on the natural numbers by taking various procedures into the transfinite. The Turing jump is iterated through he recursive ordinals to provide the skeleton of the hyperarithmetic sets. These are proven to be the $\Delta _{1}^{1}$ sets. (This is both an analog and generalization of the theorem of descriptive set theory that the Borel sets are those which are both analytic and co-analytic.) Other theorems that can be seen as results of descriptive set theory will also be proven such as every uncountable $\Sigma _{1}^{1}$ (analytic) set contains a perfect subset. The relativized relation of being hyperarithmetic (or $\Delta _{1}^{1}$) in provides the notion of hyperdegree. We will analyze the structure of the hyperdegrees and develop various appropriate notions of forcing. We will prove that the structure is rigid and that a relation on it is definable if and only if it is definable in second order arithmetic.