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Some models for intuitionistic finite type arithmetic with fan functional

Published online by Cambridge University Press:  12 March 2014

A. S. Troelstra
A. S. Troelstra*
Affiliation:
Mathematisch Instituut, Universiteit van Amsterdam, Netherlands
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Extract

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].

The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional (HAω + MUC). The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+ as defined there cannot be shown to have the required properties in EL + QF-AC, the reason being that a change in the definition of W12 alone does not suffice—if one wishes to establish closure under the operations of HAω the definitions of W1σ for other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.

For a proper understanding, we should perhaps note already here that on the assumption of the fan theorem, ECF+ as defined in [T4] and the new model of this note coincide (since then the definition of W12 [T4, p. 594] is equivalent to the definition for W12 in the case of ECF); but in EL it is impossible to prove this (and under assumption of Church's thesis the two models differ).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 2 , June 1977 , pp. 194 - 202
Copyright
Copyright © Association for Symbolic Logic 1977

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References

REFERENCES

[B]Beeson, M., Principles of continuous choice and continuity of functions in formal systems for constructive mathematics (to appear).Google Scholar
[F]Feferman, S., A language and axioms for explicit mathematics, Algebra and logic, Lecture Notes in Mathematics, vol. 450, Springer-Verlag, Berlin, 1975, pp. 87–139.Google Scholar
[K1]Kleene, S. C., Formalized recursive junctionals and formalized realizability, Memoirs of the American Mathematical Society, No. 89 (1969), Providence, R.I.Google Scholar
[L]Lifšic, V. A., Metamathematical interpretation of the fan theorem (Russian, with English summary), Zapiski naučnyh seminarov, LOMI, vol. 32 (1972), pp. 45–52.Google Scholar
[Sch]Schwichtenberg, H., Einige Anwendungen von unendlichen Termen und Wertfunktionalen, Habilitationsschrift, Westfälische Wilhelms-Universität, Münster (Westf.), 1973.Google Scholar
[T1]Troelstra, A. S. (Editor), Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, 1973.CrossRefGoogle Scholar
[T4]Troelstra, A. S., Note on the fan theorem, this Journal, vol. 39 (1974), pp. 584–596.Google Scholar
[T5]Troelstra, A. S., Computability of terms and notions of realizability for intuitionistic analysis, Report 71–02, Department of Mathematics, University of Amsterdam, 1971.Google Scholar