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Relative computability in the effective topos

Published online by Cambridge University Press:  24 October 2008

Wesley Phoa
Wesley Phoa
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
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Extract

Let ℕ be the natural numbers object in , the effective topos. It was shown in [1] that the maps ℕ → ℕ are (internally or externally) just the total recursive functions. Now a subset A ⊆ ω of natural numbers corresponds to a ¬¬-closed subobject A ↣ ℕ; let kA be the least topology forcing A to be decidable, and let ℕA be the sheafification of ℕ with respect to this topology. Then one would expect the maps ℕ → ℕA to be the total functions recursive in A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 3 , November 1989 , pp. 419 - 422
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Hyland, J. M. E.. The effective topos. In The L. E. J. Brouwer Centenary Symposium (ed. Troelstra, A. S. and Van Dalen, D.) (North-Holland, 1982).Google Scholar
[2]Hyland, J. M. E., Johnstone, P. T. and Pitts, A. M.. Tripos theory. Math. Proc. Cambridge Philos. Soc. 88 (1980), 205–231.CrossRefGoogle Scholar
[3]Hyland, J. M. E., Robinson, E. and Rosolini, G.. Discrete objects in the effective topos. (Preprint.)Google Scholar
[4]Pitts, A. M.. The theory of triposes. Ph.D. thesis, University of Cambridge (1981).Google Scholar