Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-22T14:28:15.472Z Has data issue: false hasContentIssue false
  • English
  • Français

Axiomatic recursion theory and the continuous functionals

Published online by Cambridge University Press:  12 March 2014

Simon Thompson
Simon Thompson*
Affiliation:
Computing Laboratory, University of Kent at Canterbury, Canterbury CT2 7NF, England
Get access Rights & Permissions [Opens in a new window]

Abstract

We define, in the spirit of Fenstad [2], a higher type computation theory, and show that countable recursion over the continuous functionals forms such a theory. We also discuss Hyland's proposal from [4] for a scheme with which to supplement S1–S9, and show that this augmented set of schemes fails to generate countable recursion. We make another proposal to which the methods of this section do not apply.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 2 , June 1985 , pp. 442 - 450
Copyright
Copyright © Association for Symbolic Logic 1985

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.)

References

REFERENCE

[1]Feferman, S., Inductive schemata and recursively continuous functionals, Logic Colloquium 76 (Gandy, R. O. and Hyland, J. M. E., editors), North-Holland, Amsterdam, 1977, pp. 373–392.Google Scholar
[2]Fenstad, J. E., General recursion theory: an axiomatic approach, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1980.CrossRefGoogle 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. 407–438.Google Scholar
[4]Hyland, J. M. E., The intrinsic recursion theory on the countable or continuous functionals, Generalised recursion theory. II (Fenstad, J. E., Gandy, R. O., and Sacks, G. E., editors), North-Holland, Amsterdam, 1978, pp. 135–145.Google Scholar
[5]Kechris, A. S. and Moschovakis, Y. N., Recursion in higher types, Handbook of mathematical logic (Barwise, K. J., editor), North-Holland, Amsterdam, 1977, pp. 681–737.CrossRefGoogle Scholar
[6]Moschovakis, Y. N., On the basic notions in the theory of induction, Logic, foundations of mathematics and computability theory (Butts, R. E. and Hintikka, J., editors), Part I, Reidel, Dordrecht, 1977, pp. 207–236.CrossRefGoogle Scholar
[7]Normann, D., Recursion on the countable functionals, Lecture Notes in Mathematics, vol. 811, Springer-Verlag, Berlin, 1980.CrossRefGoogle Scholar
[8]Normann, D., The continuous functionals: computations, recursions and degrees, Annals of Mathematical Logic, vol. 21 (1981), pp. 1–26.CrossRefGoogle Scholar
[9]Normann, D., External and internal algorithms on the continuous functionals, preprint, Oslo 1980, and in Patras Logic Symposion (Metakides, G., editor), North-Holland, Amsterdam, 1982, pp. 137–144.CrossRefGoogle Scholar
[10]Thompson, S. J., Recursion theories on the continuous functionals, D. Phil. Thesis, University of Oxford, Oxford, 1984.Google Scholar