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The countably based functionals

Published online by Cambridge University Press:  12 March 2014

John P. Hartley
John P. Hartley*
Affiliation:
University of Leeds, Leeds LS2 9JT, England
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Extract

In [5], Kleene extended previous notions of computations to objects of higher finite type in the maximal type-structure of functionals given by:

Tp(0) = N = the natural numbers,

Tp(n + 1) = NTp(n) = the set of total maps from Tp(n) to N.

He gave nine schemata, S1–S9, for describing algorithms for computations from a finite list of functionals, and indices to denote these algorithms. It is generally agreed that S1-S9 give a natural concept of computations in higher types.

The type-structure of countable functions, Ct(n) for n ϵ N, was first developed by Kleene [6] and Kreisel [7]. It arises from the notions of ‘constructivity’, and has been extensively studied as a domain for higher type recursion theory. Each countable functional is globally described by a countable amount of information coded in its associate, and it is determined locally by a finite amount of information about its argument. The countable functionals are well summarised in Normann [9], and treated in detail in Normann [8].

The purpose of this paper is to discuss a generalisation of the countable functionals, which we shall call the countably based functions, Cb(n) for n ϵ N. It is suggested by the notions of ‘predicativity’, in which we view N as a completed totality, and build higher types on it in a constructive manner. So we shall allow quantification over N and include application of 2E in our schemata. Each functional is determined locally by a countable amount of information about its argument, and so is globally described by a continuum of information coded in its associate, which will now be a type-2 object. This generalisation, obtained via associates, was proposed by Wainer, and seems to reflect topological properties of the countable functionals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 2 , June 1983 , pp. 458 - 474
Copyright
Copyright © Association for Symbolic Logic 1983

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References

REFERENCES

[1]Bergstra, J., The continuous functionate and 2E, Generalised recursion theory. II (Fenstad, J.E., Gandy, R.O. and Sacks, G.E., Editors), North-Holland, Amsterdam, 1978, pp. 39–54.Google Scholar
[2]Gandy, R.O. and Hyland, J.M.E., Computable and recursively countable functionals of higher type, Logic Colloquium 1976 (Gandy, R.O. and Hyland, J.M.E., Editors), North-Holland, Amsterdam, 1977, pp. 907–938.Google Scholar
[3]Grilliot, T.J., On effectively discontinuous type-2 objects, this Journal, vol. 36 (1971), pp. 245–248.Google Scholar
[4]Hyland, J.M.E., Filter spaces and continuous functionals, Annals of Mathematical Logic, vol. 16 (1979), pp. 101–143.CrossRefGoogle Scholar
[5]Kleene, S.C., Recursive functionals and quantifiers of finite types. I, Transactions of the American Mathematical Society, vol. 91 (1959), pp. 1–52.Google Scholar
[6]Kleene, S.C., Countable functionals, Constructhity in mathematics (Heyting, A., Editor), North-Holland, Amsterdam, 1959, pp. 87–100.Google Scholar
[7]Kreisel, G., Interpretation of analysis by means of functionals of finite type, Constructhity in mathematics (Heyting, A., Editor), North-Holland, Amsterdam, 1959, pp. 101–128.Google Scholar
[8]Normann, D., Recursion on the countable functionals, Lecture Notes in Mathematics, vol. 811, Springer-Verlag, Berlin and New York, 1980.Google Scholar
[9]Normann, D., The recursion theory of the continuous functionals, Recursion theory: its generalisations and applications (Drake, F.R. and Wainer, S.S., Editors), London Mathematical Society Lecture Notes, vol. 45, Cambridge University Press, 1980, pp. 171–183.CrossRefGoogle Scholar