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A system of axiomatic set theory—Part I

Published online by Cambridge University Press:  12 March 2014

Paul Bernays
Paul Bernays*
Affiliation:
Zurich
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Extract

Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).

The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.

The theory is not set up as a pure formalism, but rather in the usual manner of elementary axiom theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.

On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 2 , Issue 1 , March 1937 , pp. 65 - 77
Copyright
Copyright © Association for Symbolic Logic 1937

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References

1 This system was first introduced by the author in a lecture on “Mathematical Logic” at the University of Göttingen, 1929–30.

2 Neumann, J. V., Eine Axiomalisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154 (1925), pp. 219–240 CrossRefGoogle Scholar; Die Axiomalisierung der Mengenlehre, Mathematische Zeitschrift, vol. 27 (1928), pp. 669–752 CrossRefGoogle Scholar; Über eine Widerspruchsfreiheilsfrage in der axiomatischen Mengenlehre, Journal r. angew. Math., vol. 160 (1929), pp. 227–241 Google Scholar.

3 This elimination was first carried out, in two different ways, by Th. Skolem and A. Fraenkel. See Skolem, Th., Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Wissen-schaftliche Vorträge gehalten auf dem S. Kongress der skandinavischen Mathematiker in Helsingfors 1922, Helsingfors 1923, pp. 217–232 Google Scholar; and Fraenkel, A., Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre, Mathematische Annalen, vol. 86 (1922), pp. 230–237 CrossRefGoogle Scholar; Untersuchungen uuml;ber die Grundlagen der Mengenlehre, Mathematische Zeitschrift, vol. 22 (1925), pp. 250–273 CrossRefGoogle Scholar; Zehn Vorlesungen über die Grundlegung der Mengenlehre, Leipzig and Berlin 1927 Google Scholar; Eleitung in die Mengenlehre, 3rd. edn., Berlin 1928 Google Scholar.

4 Zermelo, E., Untersuchungen über die Grundlagen der Mengenlehre I, Mathemalische Annalen, vol. 65 (1908), pp. 261–281 CrossRefGoogle Scholar.

5 It may be observed that the von Neumann axiom system for set theory is the first example of an axiom system which is at once adequate to arithmetic and elementary in the sense just described.

6 Cf. Hilbert, and Bemays, , Grundlagen der Mathematik I, 1934, §8Google Scholar.

7 The original Zermelo system admits the existence of elements which are not sets. Zermelo insists on this point for the sake of generality. And in his recent axiomatization of set theory ( Über Grenzzahlen und Mengenbereiche, Fundamenta maihematicae, vol. 16 (1930), pp. 29–47 CrossRefGoogle Scholar) he ex-plicitly introduces Urelemente.

In the systems of Fraenkel and v. Neumann, on the other hand, it is assumed that every element is a set. This idea of avoiding elements which are not sets was apparently first suggested by P. Finsler.

Whether the one procedure or the other is preferable depends on the purpose for which the system is intended.

8 Fraenkel, A., Über die Gleichheitsbeziehung in der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 157 (1927), pp. 79–81 CrossRefGoogle Scholar.

9 Cf. Hilbert, and Bernays, , Grundlagen der Mathematik I , §8Google Scholar.

10 This manner of representing the ordered pair is due to Kuratowski, C. (Sur la notion de l'ordre dans la théorie des ensembles, Fundamenta mathematicae, vol. 2 (1921), pp. 161–171)CrossRefGoogle Scholar.

11 This inference depends on the special form of the definition which we have adopted for the ordered pair. We could avoid this dependency by taking instead of our axiom III b(l) an axiom saying that there exists a class whose elements are the pairs of the form (c, c).

12 Of course this class can be shown to exist also in other ways. For example it can be obtained as the intersection of the class of all pairs 〈c, c〉 with the class of those pairs 〈r, s〉 in which r belongs to the class of all pairs.