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On Cantor's theorem

Published online by Cambridge University Press:  12 March 2014

W. V. Quine
W. V. Quine*
Affiliation:
Harvard University
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Extract

Among various unnatural and technically burdensome effects of the theory of types, one is the unstable character of meaningfulness: a mere permutation of variables is capable of reducing a significant context to meaninglessness. Another effect, and perhaps the most conspicuous, is the systematic reduplication to which the logical constants are subjected; the calculi of classes and relations and even arithmetic lose their unity and generality, and are reproduced anew within each type. The elaborate compensatory manoeuvres which are thus made necessary are familiar to all readers of Principle, mathematica.

In a recent publication, to be cited henceforth as New foundations, I proposed an alternative course which avoids these consequences, but which would seem to offer less assurance of consistency. My efforts to derive a contradiction have delivered none, but they have lent a strange aspect to Cantor's proof that every class has more subclasses than members. This result is the topic of the present paper.

Preparatory to sketching the new system, which I shall call S′, I shall sketch a very similar but contradictory system S. By way of primitives S involves just membership, universal quantification, and alternative denial (Sheffer's stroke function), together with general variables “x”, “y”, …; the adequacy of this equipment as a basis for mathematical logic is made evident by Wiener's and Kuratowski's discovery of methods of constructing relation theory in terms of classes. Thus the formulae or statements and statement forms of S are describable recursively as follows: if a variable is put in each blank of “(ϵ)”, the result is a formula; if a variable enclosed in parentheses is prefixed to a formula, the result is a formula; and if a formula is put in each blank of “(∣)”, the result is a formula. The theorems of S are determined by a postulate and five rules, called P1 and R1-5 in New foundations. P1 and R1-3 specify various formulae as initial theorems, and R4-5 specify inferential connections for deriving further theorems. R1–2 and R4–5 are so fashioned as to provide the “theory of deduction”: they provide as theorems all those formulae which are valid by virtue merely of their structure in terms of alternative denial and quantification.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 2 , Issue 3 , September 1937 , pp. 120 - 124
Copyright
Copyright © Association for Symbolic Logic 1937

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References

1 New foundations for mathematical logic, The American mathematical monthly, vol. 44 (1937), pp. 70–80CrossRefGoogle Scholar. Acquaintance with this paper is not needed for a general understanding of the present one.

2 See Wiener, , A simplification of the logic of relations, Proceedings of the Cambridge Philosophical Society, vol. 17 (1914), pp. 387–390Google Scholar; Kuratowski, , Sur la notion de Vordere dans la théorie des ensembles, Fundamenta mathematicae, vol. 2 (1921), p. 171CrossRefGoogle Scholar; Gödel, , Über formal unenlscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 176–177Google Scholar; Tarski, Einige Betrachlungen uber die Begriffe der ω-Wider-spruchsfreiheit und der ω-Vollständigkeit, ibid., vol. 40 (1933), pp. 102-103; Quine, , Sel-theoretic foundations for logic, this Journal, vol. 1 (1936), pp. 47–49Google Scholar; New foundations, pp. 71–76.

3 See Tarski and Gödel, locis citatis. Gödel adds primitives and postulates for arithmetic, but recognizes them as superfluous.

4 In a note appended to New foundations (p. 80) after the paper was in galleys, a system still stronger than S′ was suggested. This system does prove contradictory, for in it (2) is demonstrable and the contradiction apprehended above is thus forthcoming. The decisive point is that (4), though not stratified, is nevertheless acyclic (in the sense of the cited note).

5 See Gödel, loc. cit., and Tarski, , Der Wahrheitsbegriff in den formalisierten Sprachen, Studio philosophica, vol. 1 (1935), pp. 363–364Google Scholar.