A topological analog to the Rice-Shapiro index theorem

Abstract

Ever since Craig-Beth and Addison-Kleene proved their versions of the Lusin-Suslin theorem, work in model theory and recursion theory has demonstrated the value of classical descriptive set theory as a source of ideas and inspirations. During the sixties in particular, J.W. Addison refined the technique of “conjecture by analogy” and used it to generate a substantial number of results in both model theory and recursion theory (see, e.g., Addison [1], [2], [3]).During the past 15 years, techniques and results from recursion theory and model theory have played an important role in the development of descriptive set theory. (Moschovakis's book [6] is an excellent reference, particularly for the use of recursion-theoretic tools.) The use of “conjecture by analogy” as a means of transferring ideas from model theory and recursion theory to descriptive set theory has developed more slowly. Some notable recent examples of this phenomenon are in Vaught [9], where some results in invariant descriptive set theory reflecting and extending model-theoretic results are obtained and others are left as conjectures (including a version of the well-known conjecture on the number of countable models) and in Hrbacek and Simpson [4], where a notion analogous to that of Turing reducibility is used to study Borel isomorphism types. Moschovakis [6] describes in detail an effective descriptive set theory based in large part on classical recursion theory.