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On effectively discontinuous type-2 objects

Published online by Cambridge University Press:  12 March 2014

Thomas J. Grilliot
Thomas J. Grilliot*
Affiliation:
The Pennsylvania State University, University Park, Pennsylvania 16802
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Extract

The notion of a recursive function with type-2 arguments was introduced by Kleene [4]. Soon E, the type-2 object that introduces numerical quantification, began to play a central role in the development of the theory. First, Kleene [4, XLVIII] proved that the hyperarithmetical functions are exactly the functions recursive in E. Later, Gandy [1, p. 14] proved the existence of a selection operator recursive in E (that is, some ψ(х, ), partial recursive in E, having the property that, if {х}(y, ) converges for some y, then ψ(х, ) converges and is such a y).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 2 , June 1971 , pp. 245 - 248
Copyright
Copyright © Association for Symbolic Logic 1971

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References

[1]Gandy, R. O., General recursive functionals of finite type and hierarchies of functions (mimeographed paper given at the symposium of mathematical logic held at the University of Clermont Ferrand in 06 1962).Google Scholar
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[5]Kleene, S. C., Recursive functionals and quantifiers of finite types. II, Transactions of the American Mathematical Society, vol. 108 (1963), pp. 106–142.Google Scholar
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[8]Solovay, R. M., paper presented at April 1970 meeting of Association for Symbolic Logic, abstract, this Journal, vol. 36 (1971), p. 374.Google Scholar