Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T01:08:17.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Real numbers and functions in the Kleene hierarchy and limits of recursive, rational functions

Published online by Cambridge University Press:  12 March 2014

N. Z. Shapiro
N. Z. Shapiro*
Affiliation:
The Rand Corporation
Get access Rights & Permissions [Opens in a new window]

Extract

Let ƒ be a real number. It is well known [7] that the set of rational numbers which are less than ƒ is a recursive set if and only if ƒ is representable as the limit of a recursive, recursively convergent sequence of rational numbers. In this paper we replace the condition that the set of rational numbers less than ƒ is recursive by the condition that this set is at various points in the Kleene hierarchy, and we replace the recursive, recursively convergent limit by a variety of other recursive limiting processes.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 2 , 25 July 1969 , pp. 207 - 214
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.)

References

[1] Case, Robert W., O.S.B., Partial predicates, Ph.D. thesis, Yeshiva University, New York, September 1966.Google Scholar
[2] Davis, Martin, Computability and unsohabiiity, McGraw-Hill, New York, 1958.Google Scholar
[3] Mark Gold, E., Limiting recursion, this Journal , vol. 30 (1965), pp. 28–48.Google Scholar
[4] Moschovakis, Y. N., Notation systems and recursive ordered fields, Compositio mathematica, vol. 17 (1965), pp. 40–71.Google Scholar
[5] Post, Emil L., Degrees of recursive unsohabiiity, Preliminary report, Abstract number 269, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 641–642.Google Scholar
[6] Putnam, Hilary, Trial and error predicates and the solution to a problem of Mostowski, this Journal , vol. 30 (1965), pp. 49–57.Google Scholar
[7] Rice, H. G., Recursive real numbers, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 784–791.Google Scholar
[8] Shapiro, Norman Z., Degrees of computability, Transactions of the American Mathematical Society, vol. 82 (1956), pp. 281–299.Google Scholar
[9] Shoenfield, J. R., On degrees of unsohabiiity, Annals of mathematics, vol. 69 (1959), pp. 644–653.Google Scholar