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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

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