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A relative consistency proof1

Published online by Cambridge University Press:  12 March 2014

Joseph R. Shoenfield
Joseph R. Shoenfield*
Affiliation:
Duke University
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Extract

Let C be an axiom system formalized within the first order functional calculus, and let C′ be related to C as the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. (An exact definition of C′ will be given later.) Ilse Novak [5] and Mostowski [8] have shown that, if C is consistent, then C′ is consistent. (The converse is obvious.) Mostowski has also proved the stronger result that any theorem of C′ which can be formalized in C is a theorem of C.

The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction in C from a contradiction in C′. We could, of course, obtain such a contradiction by proving the theorems of C one by one; the above result assures us that we must eventually obtain a contradiction. A similar process is necessary to obtain the proof of a theorem in C from its proof in C′. The purpose of this paper is to give a new proof of these theorems which provides a direct method of obtaining the desired contradiction or proof.

The advantage of the proof may be stated more specifically by arithmetizing the syntax of C and C′.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 19 , Issue 1 , March 1954 , pp. 21 - 28
Copyright
Copyright © Association for Symbolic Logic 1954

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Footnotes

1

The author wishes to thank the referee for various suggestions, especially concerning the advantages of the proof given here over previous proofs.

References

REFERENCES

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