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Derived rules of inference related to the continuity of effective operations

Published online by Cambridge University Press:  12 March 2014

Michael J. Beeson
Michael J. Beeson*
Affiliation:
University of Texas, Austin, Texas 78712
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Extract

In [B1] and [B2] we have shown that the classical theorems on the continuity of effective operations are not derivable in various constructive formal systems. The purpose of this paper is to establish that these systems are closed under the corresponding rules of inference, which have the form: If e can be proved to be (the index of) an effective operation, then e can be proved to be continuous. In fact we establish closure under apparently stronger rules of the form, if e can be proved to be an effective operation, then e* can be proved to be (the index of) a modulus of continuity for e, where e* is canonically obtained from e.

We again deal with two kinds of effective operations: (i) partial operations on partial arguments, and (ii) total operations on total arguments. The continuity theorem for operations of kind (i) is due to Myhill-Shepherdson [MS] and is called MS; for kind (ii) is due to Kreisel, Lacombe, and Shoenfield [KLS] and is called KLS. We also deal with effective operations on the (recursive) real numbers, which are of kind (ii). In [B1], [B2] it is shown that KLS and MS are derivable in in-tuitionistic arithmetic HA from Markov's principle MP (this follows from examination of the classical proofs); that they do not imply MP in HA; and that they are not derivable in HA. (These theorems remain true when HA is replaced by various stronger systems.) The derived rules to be established in this paper therefore represent nontrivial closure properties of the systems to which they apply.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 328 - 336
Copyright
Copyright © Association for Symbolic Logic 1976

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References

BIBLIOGRAPHY

[B1] Beeson, M., The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations, this Journal, vol. 40 (1975), pp. 321–346.Google Scholar
[B2] Beeson, M., The unprovability in intuitionistic formal systems of the continuity of effective operations on the reals (to appear).Google Scholar
[Bi] Bishop, E., Foundations of constructive analysis, McGraw-Hill, New York, 1967.Google Scholar
[KLS] Lacombe, G. Kreisel and Shoenfield, J., Partial recursive functionals and effective operations, Constructivity in mathematics (Proceedings of the Colloquium in Amsterdam, 1957) (Heyting, Editor) North-Holland, Amsterdam, 1959.Google Scholar
[MS] Myhill, J. and Shepherdson, J. C., Effective operations on partial recursive functions, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, No. 1 (1955), pp. 310–317.Google Scholar
[Tr] Troelstra, A., Notions of realizability for intuitionistic arithmetic and intuitionistic arithmetic in all finite types, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J. E., Editor), North-Holland, Amsterdam, 1971.Google Scholar
[Tr2] Troelstra, A., Metamathematical investigation of intuitionistic arithmetic and analysis, Springer, New York, 1974.Google Scholar