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The theory of classes A modification of von Neumann's system

Published online by Cambridge University Press:  12 March 2014

Raphael M. Robinson
Raphael M. Robinson*
Affiliation:
Brown University
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Extract

1. The theory of classes presented in this paper is a simplification of that presented by J. von Neumann in his paper Die Axiomatisierung der Mengenlehre. However, this paper is written so that it can be read independently of von Neumann's. The principal modifications of his system are the following.

(1) The idea of ordered pair is defined in terms of the other primitive concepts of the system. (See Axiom 4.3 below.)

(2) A much simpler proof of the well-ordering theorem, based on von Neumann's equivalence axiom (Axiom 2.2 below) is given. (More exactly, the theory of ordinal numbers, on which the proof of the well-ordering theorem is based, is simplified—see §8.)

(3) Functions are not assumed to be defined for all arguments. In place of “ = A” we have “is undefined.” This makes possible a constructive interpretation of the system. (See §2.)

2. We wish first to give a rough picture of the system which we are trying to construct. Suppose that we have the idea “class,” but no material from which to construct classes. Nevertheless, we can construct the class 0 having no elements. And then the class 1 having 0 as its only element. And then the classes {1}, and {0, 1}, etc., using always the previously constructed classes as elements. To extend this method to infinite classes, we must give rules telling what elements are to be included. Since the nature of the rules which we can use is not altogether clear, we try to formalize the whole system; that is, to set up a system of axioms which seems to characterize this system of classes. Now in this set of axioms we find it necessary to use the idea of function (in the “Axiom of Replacement”).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 2 , Issue 1 , March 1937 , pp. 29 - 36
Copyright
Copyright © Association for Symbolic Logic 1937

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References

1 Mathematische Zeitschrift, vol. 27 (1928), pp. 669–752 CrossRefGoogle Scholar. See also his paper Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 160 (1929), pp. 227–241 Google Scholar.

2 See, for example, Fraenkel, , Zehn Vorlesungen über die Grundlegung der Mengenlehre, 1927, p. 115 CrossRefGoogle Scholar. The axiom as given there reads, “Ist m eine Menge und ϕ(x) eine Funktion, so existiert auch die Menge, die aus m entsteht, falls jedes Element y von m durch ϕ(y) ersetzt wird.”

3 von Neumann, Compare, Die Axiomatisierung der Mengenlehre, p. 681 Google Scholar.

4 If we wish, the use of Axiom 7(a) may be avoided in the following theorems. The necessary changes are given in this footnote.

To the definition of ordinal number add the following condition: ON4. If F ≠ 0 and F -∈ N, then there is an x ∈ F, such thai there is no y with y ∈ x and y ∈ F. (That is, we assume that Axiom 7(a) is true for a subdomain of an ordinal number.) The deduction of the results of the first paragraph of this section can then be made for elements u; u, v; or u, v, w of an ordinal number. All the deductions in the following theorems based on Axiom 7(a) can be made equally well from ON4. In the Theorems 1.2, 3, 5.1, and 6, we must verify that the condition ON4 is satisfied by the domains there asserted to be ordinal numbers. In the first three cases this is clear, since the domains in question are sub-domains of ordinal numbers. In the last case it follows from Theorem 5.2 and ON4: By Theorem 5.2, there is a smallest ordinal M in any domain F ≠ 0 of ordinal numbers. This M satisfies the condition for x in ON4; for if there were a y with y ∈ M and y ∈ F, then we should have M -∈ y, hence y ∈ y, which is impossible.