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Generalized cohesiveness

Published online by Cambridge University Press:  12 March 2014

Tamara Hummel and
Carl G. Jockusch Jr.
Tamara Hummel
Affiliation:
Department of Mathematics, Allegheny College, 520 N. Main St., Meadville, PA 16335, USA E-mail: thummel@alleg.edu
Carl G. Jockusch Jr.
Affiliation:
Department of Mathematics, University of Illinois, 1409 W.Green St., Urbana, IL 61801, USA E-mail: jockusch@math.uiuc.edu
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Abstract

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show that for all n ≥ 2, there exists a n-cohesive set. We improve this result for n = 2 by showing that there is a 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0(n+1) and also characterize the jumps of the n-r-cohesive degrees.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999, pp. 489 - 516
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Cholak, P., Jockusch, C. G. Jr., and Slaman, T., On Ramsey's theorem for pairs, to appear.Google Scholar
[2] Cooper, S. B., Jump equivalence of the hyperhyperimmune sets, this Journal, vol. 37 (1972), pp. 598–600.Google Scholar
[3] Hummel, T., Effective versions of Ramsey's theorem, Ph.D. thesis , University of Illinois at Urbana–Champaign, Urbana, Illinois, 1993.Google Scholar
[4] Hummel, T., Effective versions of Ramsey's theorem: Avoiding the cone above 0′, this Journal, vol. 59 (1994), pp. 1301–1325.Google Scholar
[5] Jockusch, C. G. Jr., Ramsey's theorem and recursion theory, this Journal, vol. 37 (1972), pp. 268–280.Google Scholar
[6] Jockusch, C. G., Upward closure and cohesive degrees, Israel Journal of Mathematics, vol. 15 (1973), pp. 332–335.Google Scholar
[7] Jockusch, C. G. Jr. and Soare, R. I., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[8] Jockusch, C. G. Jr. and Stephan, F., A cohesive set which is not high, Mathematical Logic Quarterly, vol. 39 (1993), pp. 515–530.CrossRefGoogle Scholar
[9] Jockusch, C. G., Correction to “A cohesive set which is not high”, Mathematical Logic Quarterly, vol. 43 (1997), p. 569.Google Scholar
[10] Martin, D. A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295–310.Google Scholar
[11] Martin, D. A., Completeness, the recursion theorem, and effectively simple sets, Proceedings of the American Mathematical Society, vol. 17 (1966), pp. 838–842.Google Scholar
[12] Ramsey, F. P., On a problem informal logic, Proceedings of the London Mathematical Society, vol. 30 (1930), pp. 264–286.Google Scholar
[13] Seetapun, D. and Slaman, T., On the strength of Ramsey's theorem, Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 570–582.Google Scholar
[14] Simpson, S., Degrees ofunsolvability: a survey of results, Handbook of mathematical logic (Barwise, J., editor), North–Holland Publishing Company, Amsterdam, New York, Oxford, 1977, pp. 631–652.Google Scholar
[15] Soare, R. I., Recursively enumerable sets and degrees, Springer–Verlag, Berlin, 1987.CrossRefGoogle Scholar
[16] Soare, R. I., Computability and recursion, The Bulletin of Symbolic Logic, vol. 2 (1996), pp. 284–321.CrossRefGoogle Scholar
[17] Stephan, F., Private communication.Google Scholar