The Ramsey property and degrees in the analytical hierarchy

Citation

Kastanas, Ilias George (1981) The Ramsey property and degrees in the analytical hierarchy. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/pp79-pm74. https://resolver.caltech.edu/CaltechTHESIS:01202017-152612696

Abstract

In Chapter I we review some known results about the Ramsey theory for partitions of reals, and we present a certain two-person game such that if either player has a winning strategy then a homogeneous set for the partition can be constructed, and conversely. This gives alternative proofs of some of the known results. We then discuss possible uses of the game in obtaining effective versions and prove a theorem along these lines.

In Chapter II we study the structure of initial segments of the Δ¹_2n+1-degrees, assuming Projective Determinacy. We show that every finite distributive lattice is isomorphic to such an initial segment, and hence that the first-order theory of the ordering of Δ¹_2n+1-degrees is undecidable.

In Chapter III we extend Friedberg's Jump Inversion theorem to Q_2n+1-degrees, after noticing that it fails tor Δ¹_2n+1-degrees. We assume again Projective Determinacy.

Thesis Files