One of the principal threads in Mathematical Logic is to give a mathematical analysis of the notion of definability. Of particular interest is the relative definability of reals. Based on the fineness of ingredients that come into a particular definability notion, we obtain a whole spectrum of relative definability: effective translation (many-one reducibility), computation (Turing reducibility), existential definability (enumeration reducibility), number theoretic definability (arithmetic reducibility), and definability by the Borel operations (hyperarithmetic reducibility). We propose to study the spectrum of relative definability between subsets of the natural numbers. At the two endpoints, many-one and hyperarithmetic, we do have complete accounts and those accounts are completely different. We have only partial understanding of the spectrum between these two extremes and many important open questions. The main objective of this project is to study the to understand where in the spectrum moving from top to bottom the difference occurs. A special focus in this project will be the pair of the Turing and Enumeration degrees which is expected to hold the key for understanding this central question. The proposed project is inter-sectoral in its essence, as it examines the connection between structures that arise from Computability Theory with the structure of Second Order Arithmetic.The methodology used for these investigations place them at the intersecting point between the Computability Theory, Set Theory and Descriptive Set Theory.
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