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A system of axiomatic set theory. Part IV. General set theory38

Published online by Cambridge University Press:  12 March 2014

Paul Bernays
Paul Bernays*
Affiliation:
Zurich
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Extract

Our task in the treatment of general set theory will be to give a survey for the purpose of characterizing the different stages and the principal theorems with respect to their axiomatic requirements from the point of view of our system of axioms. The delimitation of “general set theory” which we have in view differs from that of Fraenkel's general set theory, and also from that of “standard logic” as understood by most logicians. It is adapted rather to the tendency of von Neumann's system of set theory—the von Neumann system having been the first in which the possibility appeared of separating the assumptions which are required for the conceptual formations from those which lead to the Cantor hierarchy of powers. Thus our intention is to obtain general set theory without use of the axioms V d, V c, VI.

It will also be desirable to separate those proofs which can be made without the axiom of choice, and in doing this we shall have to use the axiom V*—i.e., the theorem of replacement taken as an axiom. From V*, as we saw in §4, we can immediately derive V a and V b as theorems, and also the theorem that a function whose domain is represented by a set is itself represented by a functional set; and on the other hand V* was found to be derivable from V a and V b in combination with the axiom of choice. (These statements on deducibility are of course all on the basis of the axioms I–III.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 7 , Issue 4 , December 1942 , pp. 133 - 145
Copyright
Copyright © Association for Symbolic Logic 1943

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Footnotes

38

Parts I, II, III appeared in this Journal, vol. 2 (1937), pp. 65–77, vol. 6 (1941) pp. 1–17, and vol. 7 (1942), pp. 65–89. Part V, continuing the treatment of general set theory, will appear in a later number of this Journal.

References

39 Part II, pp. 2–3, consequence 3, consequence 4 and remark.

40 Part II, pp. 4–5, consequence 8.

41 Here and in similar cases, the basic axioms I—III are presupposed as a means of deduction without being expressly mentioned.

42 Part II, p. 4, consequence 7.

42a This kind of a derivation of Boolean algebra of course is not an independent foundation, but the fundamental operations and relations of Boolean algebra are reduced, by means of the logical concepts, to the relation of a thing (set) belonging to a class. (Added September 28, 1942.)

43 See in particular Part I, p. 76, assertions 8 and 9.

44 Concerning the proofs which have been given of this theorem see Borel, E., Leçons sur la théorie des fonctions, first edition, Paris 1898, Note I, pp. 102–107Google Scholar, and Korselt, A., Über einen Beweis des Äquivalenzsatzes, Mathematische Annalen, vol. 70 (1911), pp. 294–296.CrossRefGoogle Scholar In the proofs of Korselt, Zermelo, and Peano, the concept of finite number is eliminated by the Dedekind method of operating with intersections. However, this method of proof is in some respects less elementary, and this has the effect that it is applicable in our system only to the case that the class C (in our above formulation of the Bernstein theorem) is represented by a set.

44a The class S can be defined in other words as the class of those elements of C which, for at least one element α of C ÷ B, belong to the converse domain of the iterator of F on a—see Part II, §6, p. 12. (Added September 28, 1942.)

45 The concept of the transitive closure of a set amounts to the same thing as Finsler's concept of the system der in einer Menge wesentlichen Mengen, though the latter is defined in a somewhat different way. Finsler, P., Über die Grundlegung der Mengenlehre, Mathematische Zeitschrift, vol. 25 (1926), see §7, pp. 693–694.CrossRefGoogle Scholar

46 See Part II, §4, p. 1, consequence 2 and remark.

47 This theorem was proved by Zermelo in 1904 as a generalization of a theorem presented by Julius König at the Heidelberg Congress of 1904. Cf. König, J., Zum Kontinuum-Problem, Mathematische Annalen, vol. 60 (1905), pp. 177–180CrossRefGoogle Scholar, and Zermelo, E., Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Annalen, vol. 65 (1908), see theorem 33 and footnote, pp. 277–279.CrossRefGoogle Scholar In the following statement and its proof, as will be seen, a restricting premiss of both the König and the Zermelo theorem, which was adapted to the theory of powers, is eliminated.

48 Cf. Zermelo, E., Beweis, dass jede Menge wohlgeordnet werden kann, Mathematische Annalen, vol. 59 (1904), pp. 514–516.CrossRefGoogle Scholar

49 See Part II, §6, p. 16.

50 Cf. Sierpiński, W., Une remarque sur la notion de l'ordre, Fundamenta mathematicae, vol. 2 (1921), pp. 199–200.CrossRefGoogle Scholar The Sierpiński concept of order is a modification of that introduced by Kuratowski, in his paper, Sur la notion de l'ordre dans la théorie des ensembles, Fundamenta mathematicae, vol. 2 (1921), pp. 161–171.CrossRefGoogle Scholar The idea of representing any ordering of a set by a class of subsets such that any two distinct subsets a and b belonging to the class satisfy the condition a ⊂ b v b ⊂ a goes back to Hessenberg. Cf. Hessenberg, G., Grundbegriffe der Mengenlehre, Abhandlungen der Fries'schen Schule (Göttingen), n. s. vol. 1 (1906), in particular pp. 674–685Google Scholar (“Vollständig ordnende Systeme”).

51 Zermelo, E., Neuer Beweis für die Möglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65 (1908), see §1, pp. 107–111.CrossRefGoogle Scholar

52 Cf. Hartogs, F., Über das Problem der Wohlordnung, Mathematische Annalen, vol. 76 (1915), pp. 438–443.CrossRefGoogle Scholar