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On the consistency of Quine's New foundations for mathematical logic

Published online by Cambridge University Press:  12 March 2014

Barkley Rosser
Barkley Rosser*
Affiliation:
Cornell University
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Extract

In the present paper, the various means used by the author in attempting to find a contradiction in Quine's New foundations are sketched, and the reason why each method failed is indicated. It was not Quine's system itself which was tested, but a stronger one obtained by adding a rule of Kleene's type to Quine's system. Reasons are presented why a contradiction in the stronger system should be as damning to Quine's system as a contradiction in Quine's system itself. Two other features of the system are worthy of note. One is the fact that only two symbols are used in building up the formulas of the system. Other systems have been built using only two symbols, but the particular method used in this paper is very flexible and simple, and is peculiarly adapted to the use of the Gödel technique. It was suggested by the consideration that the formulas of any system can be written by the use of only two symbols by first assigning Gödel numbers to the formulas and then writing those numbers in the binary scale of notation. The second feature is that ι (to be used in ιxp, meaning “the x such that p”) is an explicit and integral part of the system, and the axioms and rules governing its use are such as to make it very simple to handle. By a process similar to “Die Eliminierbarkeit der ι-Symbole” of Hilbert-Bernays, it is shown that the system involving the ι can be reduced to one not involving it. The points of difference with the Hilbert-Bernays technique make possible an especially unhampered use of ι.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 4 , Issue 1 , March 1939 , pp. 15 - 24
Copyright
Copyright © Association for Symbolic Logic 1939

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References

1 Quine, W. V., New foundations for mathematical logic, The American mathematical monthly, vol. 44 (1937), pp. 70–80CrossRefGoogle Scholar.

2 Rosser, Barkley, Gödel theorems for non-constructive logics, this Journal., vol. 2 (1937), pp. 129–137Google Scholar. See Rule K Ω and footnote 11, p. 134.

3 See Schönfinkel, Moses, Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92 (1924), pp. 305–316CrossRefGoogle Scholar. On p. 315, Schönfinkel suggests building a logic out of J, (, and ). As the positions of the )'s are uniquely determined by the positions of the ('s (this was noted by the Polish logicians), we have a way of building a logic out of J and (. For a logic built entirely of the symbols a and b, see Quine, W. V., A theory of classes presupposing no canons of type, Proceedings of the National Academy of Sciences, vol. 22 (1936), pp. 320–326CrossRefGoogle ScholarPubMed. Chwistek has also built a logic out of two symbols. See the review of Grantee nauki by Perkal in this Journal, vol. 2, pp. 140–141. In this review, Perkal credits Tarski with a suggestion for building a logic with a single symbol.

4 Hilbert, D. and Bernays, P., Grundlagen der Mathematik, Berlin 1934Google Scholar. See pp. 422–457.

5 Kleene, S. C., A note on recursive functions, Bulletin of the American Mathematical Society, vol. 42 (1936), pp. 544–546CrossRefGoogle Scholar.

6 See Rosser, loc. cit., pp. 133–134.

7 Quine, W. V., On Cantor's theorem, this Journal, vol. 2 (1937), pp. 120–124Google Scholar.

8 Kleene, S. C. and Rosser, J. B., The inconsistency of certain formal logics, Annals of mathematics, vol. 36 (1935), pp. 630–636CrossRefGoogle Scholar.