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Real numbers and functions in the Kleene hierarchy and limits ofrecursive, rational functions

Published online by Cambridge University Press:  12 March 2014

N. Z. Shapiro
N. Z. Shapiro*
Affiliation:
The Rand Corporation
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Extract

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

Information

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

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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