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Definability, Automorphisms, and Dynamic Properties of Computably Enumerable Sets

Published online by Cambridge University Press:  15 January 2014

Leo Harrington  and
Robert I. Soare
Leo Harrington
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720-8204, USA.E-mail:[email protected]
Robert I. Soare
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1546, USA.E-mail:[email protected], http://www.cs.uchicago.edu/~soare, Papers posted at, ftp:cs.uchicago.edu/ftp/pub/users/soare
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Abstract

We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility).

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 2 , Issue 2 , June 1996 , pp. 199 - 213
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

[1] Chülak, P., Downey, R., and Stob, M., Automorphisms of the lattice of recursively enumerable sets: Promptly simple sets, Transactions of the American Mathematical Society, vol. 332 (1992), pp. 555–570.CrossRefGoogle Scholar
[2] Cholak, P. A., Automorphisms of the lattice of recursively enumerable sets, Memoirs of the American Mathematical Society, vol. 113 (1995).CrossRefGoogle Scholar
[3] Downey, R. and Stob, M., Automorphisms of the lattice of recursively enumerable sets: Orbits, Advances in Mathematics, vol. 92 (1992), pp. 237–265.CrossRefGoogle Scholar
[4] Harrington, L. and Soare, R. I., Codable sets and orbits of computably enumerable sets, Journal of Symbolic Logic , to appear.Google Scholar
[5] Harrington, L., Definable properties of the computably enumerable sets, in preparation.Google Scholar
[6] Harrington, L., Martins invariance conjecture and low sets, in preparation.Google Scholar
[7] Harrington, L., The -automorphism method and noninvariant classes of degrees, Journal of the American Mathematical Society , to appear.Google Scholar
[8] Harrington, L., Post's program and incomplete recursively enumerable sets, Proceedings of the National Academy of Sciences of the United States of America, vol. 88 (1991), pp. 10242–10246.CrossRefGoogle ScholarPubMed
[9] Harrington, L., Dynamic properties of computably enumerable sets, Computability, enumerability, unsolvability: Directions in recursion theory (Cooper, S. B., Slaman, T. A., and Wainer, S. W., editors), Proceedings of the Recursion Theory Conference, University of Leeds, Leeds, 07, 1994, London Mathematical Society Lecture Notes Series, Cambridge University Press, 1996.Google Scholar
[10] Soare, R. I., Computability and enumerability, Proceedings of the 10th International Congress for Logic, Methodology and the Philosophy of Science, Section 3: Recursion Theory and Constructivism, Florence, August 19–25, 1995 , to appear.Google Scholar
[11] Soare, R. I., The structure of computably enumerable objects as sets, Handbook of computability theory, North-Holland, Amsterdam, in preparation.Google Scholar
[12] Soare, R. I., Recursively enumerable sets and degrees: A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[13] Soare, R. I., Computability and recursion, to appear, 1996.CrossRefGoogle Scholar