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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). (shrink)

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of (...) infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined. (shrink)

ZusammenfassungAn Hand der Betrachtung einiger Hauptzüge in der neueren Entwicklung der Philosophie des logischen Empirismus und nahestehender Richtungen wird dargelegt, wie die Korrektur der zu simplifizierenden Thesen in dem ursprünglichen Programm der Wiener Schule auf eine Auseinandersetzung mit den traditionellen erkenntnistheoretischen Problemen zurückführt. — P.B.RésuméCe travail prend en considération certains développements récents de l'empirisme logique et des points de vues apparentés. Ces développements tendent sinon à un abandon, du moins à une revision de certaines positions par trop simplificatrices, tout d'abord (...) occupées par l'Ecole de Vienne. Or, ce faisant, on se trouve amené à rendre leur sens plein aux questions philosophiques qu'on pensait avoir pu écarter.Newer trends of development in the schools of logical empirism are considered. It is shown how, by the correction of the original too simplifying theses of the Vienna school, one becomes reduced to the study of the traditional problems of the philosophy of science. — P. B. (shrink)