Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-20T18:57:07.872Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivalence of some definitions of recursion in a higher type object1

Published online by Cambridge University Press:  12 March 2014

F. Lowenthal
F. Lowenthal*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, Vrije Universiteit Brussel, Brussels, Belgium
Get access Rights & Permissions [Opens in a new window]

Extract

In [4] Kleene gave a definition of recursive functionals of finite type. Later Sacks [5] and Harrington [2] gave definitions of recursion in normal functionals of finite type. These definitions, that Sacks and Harrington assumed equivalent as far as normal objects are concerned, are nevertheless very different: Kleene's definition is given in terms of an inductive definition; Sacks uses simultaneously a hierarchy (the S σ F's) and induction on the ordinals and on the type; Harrington's universe does not use the induction on the type but uses a hierarchy as Shoenfield [6]. In this paper we prove in detail that, as was expected, the three definitions are equivalent.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 427 - 435
Copyright
Copyright © Association for Symbolic Logic 1976

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

The results in this paper are contained in the author's doctoral dissertation written under the supervision of G. E. Sacks and partially supported by a B.A.E.F. grant.

References

REFERENCES

[1] Grilliot, T., Selection functions for recursive functionals, Notre Dame Journal of Formal Logic, vol. 10 (1969), pp. 225–234.Google Scholar
[2] Harrington, L., Contributions to recursion theory on higher types, Ph.D. Thesis, M.I.T., 1973.Google Scholar
[3] Harrington, L. and MacQueen, D., Selection in abstract recursion theory (in print).CrossRefGoogle Scholar
[4] Kleene, S., Recursive functionals and quantifiers of finite type. I, II, Transactions of the American Mathematical Society, vol. 91 (1959), pp. 1–52; vol. 108 (1963), pp. 106–142.Google Scholar
[5] Sacks, G., The 1-section of a type n object, Proceedings of the 1972 Oslo Colloquium in Generalized Recursion Theory, North-Holland, Amsterdam, 1974, pp. 81–96.Google Scholar
[6] Shoenfield, J., A hierarchy based on a type 2 object, Transactions of the American Mathematical Society, vol. 134 (1968), pp. 103–108.CrossRefGoogle Scholar