Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-12-05T02:36:02.190Z Has data issue: false hasContentIssue false
  • English
  • Français

A relative consistency proof1

Published online by Cambridge University Press:  12 March 2014

Joseph R. Shoenfield
Joseph R. Shoenfield*
Affiliation:
Duke University
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

[1]Church, Alonzo, Introduction to mathematical logic, Part I, Annals of Mathematics studies, no. 13, Princeton University Press, Princeton, N.J., 1944, 118 pp.Google Scholar
[2]Gödel, Kurt, The consistency of the continuum hypothesis, Annals of Mathematics studies, no. 3, Princeton University Press, Princeton, N.J., 1940, 74 pp.CrossRefGoogle Scholar
[3]Hilbert, D. and Bernays, P., Grundlagen der Mathematik, vol. 1, Berlin (Springer) 1943, xii + 471 pp.Google Scholar
[4]Hilbert, D. and Bernays, P., Grundlagen der Mathematik, vol. 2, 1939, xii + 498 pp.Google Scholar
[5]Novak, I. L., Models of consistent systems, Fundamenta mathematica, vol. 37 (1950), pp. 87–110.CrossRefGoogle Scholar
[6]Quine, W. V., New foundations for mathematical logic, The American mathematical monthly, vol. 44 (1937), pp. 70–80.CrossRefGoogle Scholar
[7]Quine, W. V., Mathematical logic, revised third printing, Harvard University Press, Cambridge, Mass., 1951.CrossRefGoogle Scholar
[8]Rosser, J. B. and Wang, Hao, Non-standard models for formal logics, this Journal, vol. 15 (1950), pp. 113–129.Google Scholar
[9]Wang, Hao, On Zermelo's and von Neumann's axioms for set theory, Proceedings of the National Academy of Sciences, vol. 35 (1949), pp. 150–155.CrossRefGoogle ScholarPubMed