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A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis28

Published online by Cambridge University Press:  12 March 2014

Paul Bernays
Paul Bernays*
Affiliation:
Zurich
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Extract

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.

We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 7 , Issue 2 , June 1942 , pp. 65 - 89
Copyright
Copyright © Association for Symbolic Logic 1942

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Footnotes

28

Part II appeared in this Journal, vol. 6 (1941), pp; 1–17. Part I appeared in vol. 2 (1937), pp. 65–77.

References

29 Zermelo, E., Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65 (1908), “Axiom VII,” pp. 266–267.Google Scholar

30 For typographical reasons we now use the character ⊆ instead of the character which was introduced in Part I, p. 66.

31 Fraenkel, A., Zehn Vorlesungen über die Grundlegung der Mengenlehre, Leipzig and Berlin 1927, “Axiom VII c,” p. 114.Google ScholarRobinson, R. M., The theory of classes, a modification of von Neumann's system, this Journal, vol. 2 (1937), axiom 8.3, p. 31.Google Scholar In Robinson's formulation of the axiom, as in ours following, the express assertion contained in Fraenke's axiom VII c that there exists at least one set is omitted.

32 von Neumann, J., Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154 (1925), axiom V 1, p. 226.Google Scholar

33 This added remark received August 11, 1941. Editor.

34 Cf. Part II, pp. 2–3.

35 See for example von Neumann, J., Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Mathematische Annalen, vol. 102 (1929–1930), pp. 49–131Google Scholar, in particular Kap. I and Anhang I therein.

36 The remainder of Part III, from this point on, is an addition received April 25, 1941. Editor.

37 The rôle of the axiom of choice in the theory of the Cantor second number class was generally investigated by Church, Alonzo in his dissertation, Alternatives to Zermelo's assumption, Transactions of the American Mathematical Society, vol. 29 (1927), pp. 178–208.Google Scholar