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Element and number

Published online by Cambridge University Press:  22 April 2025

W.V. Quine
W.V. Quine*
Affiliation:
New York University
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Extract

Rosser pointed out recently that the system of my Mathematical logic, when curtailed by leaving out the elementhood principle *200, would hold true even of a universe wherein the only entity is the null class. In other words, all theorems of the curtailed system remain true when all atomic formulae ⌜α∊β⌝ are construed as false for all values of their variables. This observation constitutes a consistency proof of that curtailed system, in an extraordinarily strict sense; for it shows that every theorem of that curtailed system is of a very special kind such that if all its quantifiers be simply rubbed out and all its atomic formulae be marked ‘F’ then the whole will receive a “T” under the ordinary truth-table computation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 6 , Issue 4 , December 1941 , pp. 135 - 149
Copyright
Copyright © Association for Symbolic Logic 1941

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References

1 Barkley Rosser, The independence of Quine's axioms *200 and *201, this JOURNAL, vol. 6 (1941), pp. 96-97.

2 This JOURNAL, vol. 6 (1941), p. 163.

2a Barkley Rosser, The Burali-Forti paradox, forthcoming in this JOURNAL.

3 J. von Neumann, Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154 (1925), pp. 219-240; vol. 155 (1926), p. 128.

4 W. V. Quine, New foundations for mathematical logic, The American Mathematical monthly, vol. 44 (1937), pp. 70-80.

4a Op. cit., second proof of †841. Note that the A of Rosser's first proof of †841 likewise has an unstratified defining condition.

5 The sense of stratification concerned here is more elaborate than in New foundations or Mathematical logic, since in Principia relations are not reduced to classes.

6 Whitehead and Russell, Principia mathematica, vol. 3, pp. 75, 80.

7 For an account of this bundle of paradoxes see §2 of my paper On the theory of types, this JOUBNAL, vol. 3 (1938), pp. 125-139.

8 It is of the latter system that my cited paper On the theory of types treats. Thus the view, accepted in that paper, that the mere requirement of stratification suffices for the avoidance of paradoxes, is in no way controverted by the present remarks.

9 Concerning the solution of Cantor's paradox for the system of New foundations see my paper On Cantor's theorem, this JOURNAL, vol. 2 (1937), pp. 120-124.

10 See Principia, vol. 2, p. 31.

11 I shall use italics to distinguish newly applied reference numbers from reference numbers whose application remains as in the book.

12 This and its sequel are called †410a ff. rather than †402 ff. in view of Mathematical logic, p. 90n. But note still that †410 is to be understood as coming after all these.

13 Von Neumann has construed the finite ordinals in this way. See below, §6.

14 The identification of ‘≤’ with ‘⊂’ is suited to the arithmetic not only of natural numbers but of cardinals and ordinals generally (cf. §6 below), and even of real numbers— both when real numbers are taken as Dedekind segment-classes of ratios (cf. Mathematical logic, pp. 271 f.) and when they are taken rather as relations Є“x such that a; is a Dedekind segment-class of ratios (cf. Principia mathematica, vol. 3, p. 336). The identification of ‘≤’ with ‘⊂’ is suited also to the arithmetic of ratios, if we merely reconstrue the ratios cumulatively; i.e., if in the first definition of §50 of Mathematical logic we change ‘ = ‘ to ‘ £'. This cumulative version of ratios is advantageous also on other grounds; for it causes the ratios to be identical with the rational reals, once the relational version of reals mentioned above is adopted. The reals come to constitute a genuine extension of the ratios.

15 Cf. Mathematical logic, p. 135.

16 Cf. pp. 204 ff., op. cit.

17 Op. cit., p. 207.

18 Op. cit., p. 205.

19 Substantially this principle is mentioned on p. 227n, op. cit. But note the misprint: there should be a dot on the last ‘ζ’.

20 Such is von Neumann's way of construing the ordinals in his paper Zur Einführung der transfiniten Zahlen, Ada litterarum ac scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum, vol. 1 (1923), pp. 199-208.