Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-27T01:00:57.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniformly introreducible sets1

Published online by Cambridge University Press:  12 March 2014

Carl G. Jockusch Jr.
Carl G. Jockusch Jr.*
Affiliation:
Northeastern University and Massachusetts Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Extract

Retraceable sets were denned in Dekker and Myhill [5]. In that paper it was pointed out that if A is a retraceable set, then A is recursive in each of its infinite subsets, i.e. A is introreducible. We would like to know to what extent theorems on retraceable sets also hold for introreducible sets; however, we have made very little progress in this direction. On the other hand, “uniformly introreducible” sets (which will be defined in §2) are more tractable, and it turns out that some results on retraceable sets extend to uniformly introreducible sets while others fail.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 4 , January 1969 , pp. 521 - 536
Copyright
Copyright © Association for Symbolic Logic 1969

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

This research was supported in part by National Science Foundation Grant GP 4361 at M.I.T. The author is grateful to the referee and T. G. McLaughlin for helpful suggestions.

References

[1]Appel, K. I., No recursively enumerable set is the union of finitely many immune retraceable sets, Proceedings of the American Mathematical Society, vol. 18 (1967), pp. 279–281.CrossRefGoogle Scholar
[2]Appel, K. I. and McLaughlin, T. G., On properties of regressive sets, Transactions of the American Mathematical Society, vol. 115 (1965), pp. 83–93.CrossRefGoogle Scholar
[3]Dekker, J. C. E., Infinite series of isols, Proceedings of symposia in pure mathematics, Vol. 5, Amer. Math. Soc., Providence, R.I., 1962, pp. 77–96.CrossRefGoogle Scholar
[4]Dekker, J. C. E., The minimum of two regressive isols, Mathematische Zeitschrift, vol. 83 (1964), pp. 345–366.CrossRefGoogle Scholar
[5]Dekker, J. C. E. and Myhill, J., Retraceable sets, Canadian journal of mathematics, vol. 10 (1958), pp. 357–373.CrossRefGoogle Scholar
[6]Grzegorczyk, A., Mostowski, A., and Ryll-Nardzewski, C., The classical and the ω-complete arithmetic, this Journal, vol. 23 (1958), pp. 188–206.Google Scholar
[7]Jockuschi, C. G., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420–436.CrossRefGoogle Scholar
[8]McLaughlin, T. G., Splitting and decomposition by regressive sets, Michigan mathematical journal, vol. 12 (1965), pp. 499–505.CrossRefGoogle Scholar
[9]McLaughlin, T. G., Splitting and decomposition by regressive sets. II, Canadian journal of mathematics, vol. 19 (1967), pp. 291–311.CrossRefGoogle Scholar
[10]McLaughlin, T. G., Some remarks on extensibility, confluence of paths, branching properties, and index sets, for certain recursively enumerable graphs, Illinois journal of mathematics, vol. 11 (1967), pp. 257–279.CrossRefGoogle Scholar
[11]Miller, W., Sets of integers and degrees of unsolvability, Master's thesis, University of Washington, Seattle, Wash, 1967.Google Scholar
[12]Myhill, J., Finitely representable functions, Constructivity in mathematics, edited by Heyting, A., North-Holland, Amsterdam, 1959, pp. 195–207.Google Scholar
[13]Rogers, H. Jr., Recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[14]Sacks, G. E., Degrees of unsolvability, Annals of mathematics, Studies No. 55, Princeton Univ. Press, Princeton, N.J., 1963.Google Scholar
[15]Spector, C., Inductively defined sets of natural numbers, Infinitistic methods, Proceedings of the Symposium of Foundations of Mathematics, Warsaw, 1961.Google Scholar
[16]Tennenbaum, S., Degrees of unsohability and the rate of growth of functions, Abstract 587–128, Notices of the American Mathematical Society, vol. 8 (1961), p. 608.Google Scholar
[17]Yates, C. E. M., Recursively enumerable sets and retracing functions, Zeitschrift für Mathematische Logik und Grandlagen der Mathematik, vol. 8 (1962), pp. 331–345.CrossRefGoogle Scholar
[18]Yates, C. E. M., Three theorems on the degrees of recursively enumerable sets, Duke mathematical journal, vol. 32 (1965), pp. 461–468.CrossRefGoogle Scholar
[19]Yates, C. E. M., Arithmetical sets and retracing functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 13 (1967), pp. 193–204.CrossRefGoogle Scholar