1937/03/01 by Paul Bernays · 5 citations
Psychology · Arts and Humanities · Computer Science · Mathematics · #Philosophy and Theoretical Science #Philosophy, Science, and History #Logic, Reasoning, and Knowledge #Axiom #Von Neumann architecture #Mathematics #Constructive set theory #Axiomatic system #Zermelo–Fraenkel set theory #Rule of inference #Set theory #Axiom of choice #Inference #Calculus (dental) #Set (abstract data type) #Mathematical economics #Algebra over a field #Computer science #Pure mathematics #Artificial intelligence
paper · doi:10.2307/2268862
openalex publication_date 1937/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
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).