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A system of axiomatic set theory—Part II13

Published online by Cambridge University Press:  12 March 2014

Paul Bernays
Paul Bernays*
Affiliation:
Zurich
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Extract

For the formulation of the remaining axioms we need the notions of a function and of a one-to-one correspondence.

We define a function to be a class of pairs in which different elements always have different first members; or, in other words, a class F of pairs such that, to every element a of its domain there is a unique element b of its converse domain determined by the condition 〈a, b〉ηF. We shall call the set b so determined the value of F for a, and denote it (following the mathematical usage) by F(a).

A set which represents a function—i.e., a set of pairs in which different elements always have different first members—will be called a functional set.

If b is the value of the function F for a, we shall say that F assigns the set b to the set a; and if a functional set f represents F, we shall say also that f assigns the set b to the set a.

A class of pairs will be called a one-to-one correspondence if both it and its converse class are functions. We shall say that there exists a one-to-one correspondence between the classes A and B (or of A to B) if A and B are domain and converse domain of a one-to-one correspondence. Likewise we shall say that there exists a one-to-one correspondence between the sets a and b (or of a to b) if a and b respectively represent the domain and the converse domain of a one-to-one correspondence. In the same fashion we speak of a one-to-one correspondence between a class and a set, or a set and a class.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 6 , Issue 1 , March 1941 , pp. 1 - 17
Copyright
Copyright © Association for Symbolic Logic 1941

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Footnotes

13

Part I appeared in this Journal, vol. 2 (1937), pp. 65–77.

References

14 The axioms V a, c, d correspond respectively to Zermelo's Axiom der Aussonderung, Axiom der Vereinigung, and Axiom der Potenzmenge. Cf. Zermelo, E., Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65 (1908), pp. 261–281CrossRefGoogle Scholar. Axiom V b is a modification of Fraenkel's Axiom der Ersetzung. Cf. Fraenkel, A., Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre, Mathematische Annalen, vol. 86 (1922), pp. 230–237CrossRefGoogle Scholar; and Zehn Vorlesungen über die Grundlegung der Mengenlehre, Leipzig and Berlin 1927, see p. 115 and pp. 104–110Google Scholar.

15 This is the assertion of the multiplicative axiom, which was first introduced by Russell (in his paper On some difficulties in the theory of transfinite numbers and order types, Proceedings of the London Mathematical Society, ser. 2 vol. 4 (1906–1907), see pp. 47–52Google Scholar) as a modification of the original Zermelo choice postulate (cf. Beweis, dass jede Menge wohlgeordnet werden kann, Mathematische Annalen, vol. 59 (1904), see p. 516)Google Scholar, and was independently stated by Zermelo as axiom of choice, from which he was able to infer, within his axiom system, the assertion of his former postulate (cf. Neuer Beweis für die Möglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65 (1908), see p. 110Google Scholar, and Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65 (1908), see p. 266 and pp. 273–274)Google Scholar.

16 Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre, Mathematische Annalen, vol. 86 (1922), pp. 230–237, see p. 233CrossRefGoogle Scholar.

17 Cf. Dedekind, Richard, Was sind und was sollen die Zahlen?, Braunschweig 1888, §5Google Scholar. —Of course Dedekind makes no distinction between classes and sets; as a matter of fact he uses neither word, but speaks of systems.

18 The idea of this axiom was first conceived by von Neumann in his paper Eine Axiomatisierung der Mengenlehre (Journal für die reine und angewandte Mathematik, vol. 154 (1925), see p. 239, axiom VI 4)Google Scholar. From the original form of it he passed to the present form of the axiom, or to a somewhat stronger axiom in a form adapted to his system, in Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre (Journal für die reine und angewandte Mathematik, vol. 160 (1929), see p. 231)Google Scholar. Zermelo independently introduced this axiom, calling it Axiom der Fundierung, in his paper Über Grenzzahlen und Mengenbereiche (Fundamenta mathematicae, vol. 16 (1930), see p. 31)Google Scholar.

19 This dependency was stated by Gödel.

20 The theory of classes, a modification of von Neumann's system, this Journal, vol. 2 (1937), pp. 29–36Google Scholar.

21 Loc. cit. (footnote 20).

22 We here modify Zermelo's definition slightly, replacing his condition “0ϵn” by “0 = n or 0ϵn,” in order that the null set may be counted amongst the ordinals, as it is under our other definitions.

22 Justification of recursive definition of numerical functions, from the set-theoretic point of view, was first given by Dedekind, in Was sind und was sollen die Zahlen?, Braunschweig 1888, §9Google Scholar. We here follow his method, but with the modification that we reduce the recursive definition to the more special case of an iteration.

24 This reduction figures in the investigations of A. Church, J. B. Rosser, and S. C. Kleene on the “λ-definability” of numerical functions. See Kleene, S. C., A theory of positive integers informal logic, American journal of mathematics, vol. 57 (1935), pp. 153–173, 219–244CrossRefGoogle Scholar, especially the passage from the theorem 15 IV to 15 V, pp. 220–221.

25 Cf., e.g., Hilbert, and Bernays, , Grundlagen der Mathematik, vol. 1, pp. 401–422Google Scholar.

26 See p. 2 above.

27 Zermelo, E., Sur les ensembles finis et le principe de l'induction complète, Acta mathematica, vol. 32 (1909), pp. 185–193 (dated 1907)CrossRefGoogle Scholar; Ueber die Grundlagen der Arithmetik, Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 aprile 1908), vol. 2 (1909), pp. 8–11Google Scholar. See also Grelling, Kurt, Die Axiome der Arithmetik, dissertation Göttingen 1910Google Scholar.