Urelement

Elements are in set theory elements which themselves contain no elements. So they form a genuine portion of the elements. Elements are distinguished from individuals, since the latter are today usually equated mathematics with elements.

Formally constitute the primary elements of the class.

In this definition, the empty set is a primary element ( the only one that is a lot ) explicitly included. At other points of view see below

As Elements mathematically unspecified, default objects and things can be considered, such as apples, pears, humans, horses, etc., which can be summarized in quantities as other elements. They correspond to objects of intuition in the amount of definition in 1895 by Georg Cantor:

On this definition, Zermelo -oriented in his axiomatization of Cantor's set theory: Both the Zermelo set theory of 1907 as well as the original ZF system of 1930 set a range of things ahead that contains the quantities as real subregion and beyond other things, 1930 which he called " Elements " was; Elements do not contain these elements, because he always looked at things as element-containing quantities. Such a set theory with basic elements meets the philosophical need of a general logical language. The mathematician Abraham Fraenkel argued in 1921 for the first time for a pure set theory without additional Elements. With its substitution axiom can namely depict a lot with basic elements to a quantity without equally powerful Elements. Therefore, one arrives at the set-theoretic description of any circumstances without additional Elements. Already the first formalization of ZF set theory by Thoralf Skolem in 1929 renounced additional Elements. The school then made so that today ZF axiom systems are usually a pure ZF set theory describe (the only original element here is the empty set). The pure set theory also has the merit of simplicity, as their simpler axioms allow simpler proofs. For primary elements you need above all an attenuated extensionality, which applies only to quantities and not for Elements; formal proofs are then cumbersome as more and additional quantity conditions have to be dragged ( in other axioms ). But there are also modern quantity gauges, take into account the original elements, such as the Ackermann set theory (variant ) or the general set theory by Arnold Oberschelp that builds on a logic of classes.

Other definition

Elements are sometimes also defined as elements that are not volumes. With this selection, then separates the empty set as a primary element, but theoretically possible but proper classes are as primordial elements of what not to Zermelo's original element -Intention fits, but enables interesting forms of set theory with real genuine classes. Here, however, the primary element - concept depends on the selected amount and term of the elected amount axioms, so that here there is no easy manageable situation.

Literature ( in chronological order )

  • Zermelo, Ernst: studies on the foundations of set theory, 1907, in: Mathematische Annalen 65 (1908 ) pp. 261-281
  • Fraenkel, Adolf: Introduction to set theory. . Springer Verlag, Berlin -Heidelberg - New York, 1928 reprint: Dr. Martin Sändig oHG, Walluf 1972, ISBN 3-500-24960-4.
  • Skolem, Thoralf: About some fundamental questions of Mathematics, 1929, in: selected works in logic, Oslo, 1970, pp. 227-273
  • Zermelo, Ernst: About marginal numbers and quantity ranges, in: Fundamenta Mathematicae 16 (1930 ), pp. 29-47
  • Oberschelp, Arnold: General set theory, Mannheim, Leipzig, Vienna, Zurich, 1994
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