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The axiom of infinity in Quine's New Foundations

Published online by Cambridge University Press:  12 March 2014

J. Barkley Rosser
J. Barkley Rosser*
Affiliation:
Cornell University
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Extract

We use NF to designate the system known as Quine's New Foundations (see [1] and [2]), and NF + AF to designate the same system with a suitable axiom of infinity adjoined. We use ML to designate the revised system appearing in the third printing of Quine's “Mathematical Logic” (see [3]). This system ML is just the system P proposed by Wang in [4], and essentially includes NF as a part.

The pripcipal results of the present paper are:

A. In NF the axiom of infinity is equivalent to the definability of an ordered pair having the same type as its constituents.

B. Considering the formulas of NF as having translations in the formalism of ML in the sense defined in [4], we find that a formula of NF is provable in NF if and only if its translation is provable in ML. From the fact that the axiom of infinity is provable in ML, one might be tempted to conclude that the axiom of infinity is provable in NF. It is explained why this is not a valid inference. Consequently, it would appear that there is a sense in which NF + AF is stronger than ML.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 17 , Issue 4 , December 1952 , pp. 238 - 242
Copyright
Copyright © Association for Symbolic Logic 1952

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References

REFERENCES

[1] Quine, W. V., New foundations for mathematical logic, The American mathematical monthly, vol. 44 (1937), pp. 70–80.CrossRefGoogle Scholar
[2] Rosser, Barkley, On the consistency of Quine's new foundations for mathematical logic, this Journal, vol. 4 (1939), pp. 15–24.Google Scholar
[3] Quine, W. V., Mathematical logic, revised third printing, Harvard University Press, Cambridge, Mass., 1951.CrossRefGoogle Scholar
[4] Wang, Hao, A formal system of logic, this Journal, vol. 15 (1950), pp. 25–32.Google Scholar
[5] Kuratowski, Casimir, Sur la notion de l'ordre dans la théorie des ensembles, Fundamenta mathematicae, vol. 2 (1921), pp. 161–171.CrossRefGoogle Scholar
[6] Goodman, Nelson, Sequences, this Journal, vol. 6 (1941), pp. 150–153.Google Scholar
[7] Quine, W. V., On ordered pairs, this Journal, vol. 10 (1945), pp. 95–96.Google Scholar
[8] Quine, W. V., On Cantor's theorem, this Journal, vol. 2 (1937), pp. 120–124.Google Scholar
[9] Rosser, J. Barkley and Wang, Hao, Non-standard models for formal logics, this Journal, vol. 15 (1950), pp. 113–129.Google Scholar
[10] Gödel, K., The consistency of the continuum hypothesis, Princeton University Press, 1940.Google Scholar
[11] Hailperin, T., A set of axioms for logic, this Journal, vol. 9 (1944), pp. 1–19.Google Scholar