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Isomorphisms between HEO and HROE, ECF and ICFE

Published online by Cambridge University Press:  12 March 2014

Marc Bezem*
Affiliation:
Korenbloemstraat 44, 3551 Gn Utrecht, The, Netherlands

Abstract

In this paper it will be shown that HEO and HROE are isomorphic with respect to extensional equality. This answers a question of Troelstra [T, 2.4.12, p. 128]. The main problem is to extend effective operations to a larger domain. This will be achieved by a modification of the proof of the continuity of effective operations. Following a suggestion of A. S. Troelstra, similar results were obtained for ECF(U) and ICFE(U), where U is any universe of functions closed under “recursive in”.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1985

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References

[H] Heyting, A. (editor), Constructivity in mathematics, North-Holland, Amsterdam, 1959.Google Scholar
[Kl] Kleene, S. C., Countable functionals, [H, pp. 81100].Google Scholar
[Kr] Kreisel, G., Interpretation of analysis by means of constructive functionals of finite type, [H, pp. 101–128].Google Scholar
[KLS] Kreisel, G., Lacombe, D. and Shoenfield, J. R., Partial recursive functionals and effective operations, [H, pp. 290–297].Google Scholar
[T] Troelstra, A. S., Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, 1973.Google Scholar
[Z] Zucker, J. I., Proof-theoretic studies of iterated inductive definitions and subsystems of analysis, Ph.D. Thesis, Stanford University, Stanford, California, 1971.Google Scholar
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