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A topological analog to the Rice-Shapiro index theorem

Published online by Cambridge University Press:  12 March 2014

Louise Hay  and
Douglas Miller
Louise Hay
Affiliation:
University of Illinoisat Chicago Circle, Chicago, Illinois 60680
Douglas Miller
Affiliation:
University of Illinoisat Chicago Circle, Chicago, Illinois 60680
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 4 , December 1982 , pp. 824 - 832
Copyright
Copyright © Association for Symbolic Logic 1982

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References

BIBLIOGRAPHY

[1]Addison, J.W., The theory of hierarchies, Logic, Methodology and Philosophy of Science (Proceedings of the International Congress, 1960), Stanford Unversity Press, Stanford, 1962, pp. 26–37.Google Scholar
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[5]Miller, D., Remarks on topological index sets (in preparation).Google Scholar
[6]Moschovakis, Y., Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
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[9]Vaught, R.L., Invariant sets in topology and logic, Fundamenta Mathematicae, vol. 82 (1974), pp. 269–294.CrossRefGoogle Scholar