Axiomatics of set theory without element relation
type: Item
instance of: scholarly article
title: Axiomatics of set theory without element relation
zbMATH Open document ID: 0075.00902
DOI: 10.1007/BF01350099
author: Herbert Wegel
published in: Mathematische Annalen
publication date: 1956-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/160500
review text: The author proposes an axiomatization of set theory based upon the inclusion relation. There are two sorts of variables, one for sets and one for functions. In addition to inclusion, there is only one other primitive predicate, a six-place predicate \(W(a,b,c,d,g,e)\) with the intuitive meaning that the function \(\varphi\) is applicable to the quadruple \((a,b,c,d)\) and yields the value \(e\). On the basis of thirteen axioms, the author begins the development of standard set-theoretic operations and results, reaching the theory of equivalence (one-one correspondences, equinumerous sets, etc.). The given system is shown to be imbeddable in a form of Zermelo-Fraenkel set theory, but the exact relation between the two theories is not made clear (the author claims that the membership relation is not definable in his system). The set theory based upon the given system seems extremely complicated and unwieldy, as compared to the standard systems using the membership relation.
reviewed by: Elliott Mendelson
Mathematics Subject Classification ID: 03E30
zbMATH DE Number: 3124830
zbMATH Keywords: mathematical logic and foundations
MaRDI profile type: Publication
cites work: A system of axiomatic set theory—Part I
cites work: Q5789040
cites work: Grundbegriffe einer allgemeinen Teiltheorie
cites work: Q5767692
cites work: Axiomatische Theorie der Wohlordnung. (Untersuchungen über die Grundlagen der Mengenlehre. III.)
cites work: Q4756727
description: scientific article; zbMATH DE number 3124830