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The 1-section of a countable functional

Published online by Cambridge University Press:  12 March 2014

Dag Normann and
Stan S. Wainer
Dag Normann
Affiliation:
University of Oslo, Blindern, Oslo 3, Norway
Stan S. Wainer
Affiliation:
University of Leeds, Leeds LS2 9JT, England
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Extract

The continuous or countable functionals were independently denned by Kleene [9] and Kreisel [10]. They were intended as a suitable basis for constructive mathematics, and thus it is interesting to investigate various notions of recursion on the countable functionals.

There have been two main streams in this investigation, the study of countable recursion and the study of computability or Kleene-recursion.

Countable recursion is the theory of recursion on the associates. Gandy and Hyland [3] and Hyland [7] are good sources for the recent development of countable recursion.

This paper will mostly be concerned with Kleene-recursion on the countable functionals as denned in Kleene [8] and [9]. We assume some familiarity with the countable functionals and associates, as presented in Kleene [9], Bergstra [1] or any other paper on the subject.

Pioneering work with recursion in nonnormal objects was done by Grilliot [4], who proved that a functional F of type 2 is normal if and only if its 1-section (that is the set of functions recursive in F) is closed under ordinary jump, and if and only if F is continuous on 1-section (F).

Hinman [6] constructed a countable functional that is not recursively equivalent to a function, and thereby showed that recursion in nonnormal functionals is an extension of ordinary recursion in functions.

In [6], Hinman asked if there are functionals with topless 1-sections, i.e. with no maximal elements in the semi-lattice of degrees. This was answered in the affirmative by Bergstra [1], using a spoiling construction. Thus the class of 1-sections of functionals extends the class of 1-sections of functions.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 3 , September 1980, pp. 549 - 562
Copyright
Copyright © Association for Symbolic Logic 1980

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References

REFERENCES

[1]Bergstra, J., Computability and continuity in finite types, Dissertation, Utrecht, 1976.Google Scholar
[2]Bergstra, J. and Wainer, S. S., The “real’ ordinal of the 1-section of a continuous functional, Abstract, European meeting of the Association for Symbolic, Logic, Oxford, 1976, this Journal, vol. 42 (1977), p. 440.Google Scholar
[3]Gandy, R. O. and Hyland, J. M. E., Computable and recursively countable functions of higher type, Logic Colloquium '76 (Gandy, R. O. and Hyland, J. M. E., Editors), North-Holland, Amsterdam, 1977, pp. 407–408.Google Scholar
[4]Grilliot, T., On effectively discontinuous type-2 objects, this Journal, vol. 36 (1971), pp. 245–248.Google Scholar
[5]Hinman, P., Hierarchies of effective descriptive set theory, Transactions of the American Mathematical Society, vol. 142 (1969), pp. 111–140.CrossRefGoogle Scholar
[6]Hinman, P., Degrees of continuous functionals, this Journal, vol. 38 (1973), pp. 393–395.Google Scholar
[7]Hyland, J. M. E., The intrinsic recursion theory on the countable or continuous functionals, Generalized recursion theory. II (Fenstad, J. E., Gandy, R. O. and Sacks, G. E., Editors), North-Holland, Amsterdam, 1978.Google Scholar
[8]Kleene, S. C., Recursive functionals and quantifiers of finite types. I, Transactions of the American Mathematical Society, vol. 91 (1959), pp. 1–52; II, vol. 108 (1963), pp. 106–142.Google Scholar
[9]Kleene, S. C., Countable functionals, Constructivity in mathematics (Heyting, A., Editor), North-Holland, Amsterdam, 1959, pp. 81–100.Google Scholar
[10]Kreisel, G., Interpretation of analysis by means offunctionals offinite type, Constructivity in mathematics (Heyting, A., Editor), North-Holland, Amsterdam, 1959, pp. 100–128.Google Scholar
[11]Lachlan, A. H., Uniform enumeration operations, this Journal, vol. 40 (1975), pp. 401–409.Google Scholar
[12]Moschovakis, Y. N., Hyperanalytic predicates, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 249–282.CrossRefGoogle Scholar
[13]Normann, D., A continuous functional with noncollapsing hierarchy, this Journal, vol. 43 (1978), pp. 487–491.Google Scholar
[14]Normann, D., Countable functionals and the analytic hierarchy, Oslo Preprint Series, no. 17 (1977).Google Scholar
[15]Shoenfield, J. R., A hierarchy based on a type-2 object, Transactions of the American Mathematical Society, vol. 134 (1968), pp. 103–108.CrossRefGoogle Scholar
[16]Wainer, S. S., A hierarchy for the 1-section of any type-2 object, this Journal, vol. 39 (1974), pp. 88–94.Google Scholar
[17]Wainer, S. S., The 1-section of a non-normal type-2 object, Generalized recursion theory. II (Fenstad, J. E., Gandy, R. O. and Sacks, G. E., Editors), North-Holland, Amsterdam, 1978.Google Scholar