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Note on the fan theorem

Published online by Cambridge University Press:  12 March 2014

A. S. Troelstra
A. S. Troelstra*
Affiliation:
Mathematical Institute, Oxford, England
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Extract

The principal aim of this paper is to establish a theorem stating, roughly, that the addition of the fan theorem and the. continuity schema to an intuitionistic system of elementary analysis results in a conservative extension with respect to arithmetical statements.

The result implies that the consistency of first order arithmetic cannot be proved by use of the fan theorem, in addition to standard elementary methods—although it was the opposite assumption which led Gentzen to withdraw the first version of his consistency proof for arithmetic (see [B]).

We must presuppose acquaintance with notation and principal results of [K, T], and with §1.6, Chapter II, and Chapter III, §4-6 of [T1]. In one respect we shall deviate from the notation in [K, T]: We shall use (n)x (instead of g(n, x)) to indicate the xth component of the sequence coded by n, if x < lth(n), 0 otherwise.

We also introduce abbreviations n ≤* m, a ≤ b which will be used frequently below:

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 3 , September 1974 , pp. 584 - 596
Copyright
Copyright © Association for Symbolic Logic 1974

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References

BIBLIOGRAPHY

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