The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...

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$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.

We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular--- the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals

xperience, but simply as irreducible points comparable, epistemologically, to the gods of Homer.") Of course, the theory itself does indicate di#culties in substantiating the Turing model, but, if not overstretched (viz. the ubiquitous Godel's [15], [16] Theorem) such asymptotic representations can be useful and productive adjuncts to subjective intuition. For instance, unlike in mathematics where small variations in axioms can lead to fundamentally di#erent theories, Turing nonrigidity and known countable automorphism bases indicate that although diverse basic assumptions about the real world, related to culture or religion, for example, are inevitable (perhaps even necessary), relative to the Turing model there is a convergence at higher levels of the informational structure suggested by relative rigidity of substructures. The purpose of this note is to describe how, at a more basic level, the material Universe can be modelled according to the underlying structure of