Primal element

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In set theory , primitive elements are elements that themselves do not contain any elements. So they form a real part of the elements. Primordial elements are to be distinguished from individuals , since the latter are mostly equated with elements in mathematics today.

Formally, the primary elements form the class .

In this definition, the empty set is expressly included as the original element (the only one that is a set). For other perspectives see below

As additional primitive elements (in addition to the empty set), mathematically unspecified, given objects and things can be understood, such as apples, pears, people, horses, etc., which, like other elements, can be summarized in quantities. They correspond to the objects of perception in the set definition of 1895 by Georg Cantor :

By a “set” we mean any combination M of certain well-differentiated objects m of our perception or our thinking (which are called the “elements” of M) into a whole.

Zermelo oriented himself to this definition in his axiomatization of Cantor's set theory: Both the Zermelo set theory of 1907 and the original ZF system of 1930 presuppose a range of things that contains the sets as a real sub-range and also other things, which he gave the name "Urelemente" in 1930; these primordial elements contain no elements, since he always viewed things containing elements as sets. Such a set theory with additional primitive elements meets the philosophical need for a general logical language and aims at an application of set theory in other disciplines. In 1921, mathematician Abraham Fraenkel first advocated pure set theory without such additional primitive elements. With his substitution axiom, you can map a set with such 'real' primitive elements to a set of equal power without such primitive elements. Therefore one gets along with the set-theoretical description of any facts without additional primitive elements. Thoralf Skolem's first formalization of ZF set theory in 1929 already dispensed with additional primitive elements. This then caught on, so that today's ZF axiom systems usually describe a pure ZF set theory (the only original element here is the - indispensable - empty set). Pure set theory also has the advantage of simplicity, since its simpler axioms allow simpler proofs. For additional primitive elements, one needs above all a weakened extensionality , which is only valid for sets and not for “real” primitive elements; formal proofs then become more laborious, since additional set conditions (also with other axioms) have to be dragged along. But there are also modern set theories that take into account primitive elements, such as Arnold Oberschelp's general set theory , which is based on class logic .

Other definition

Sometimes primordial elements are also defined in the narrower sense as elements that are not sets. With this definition, the empty set is eliminated as a primitive element, but theoretically real classes are then possible as primitive elements. This way of thinking does not fit with Zermelo's original element intention, but enables interesting forms of set theory with real real classes. Here, however, the primitive element concept depends on the selected set concept and on the selected set axioms, so that there is no easily comprehensible factual situation here. One example is a variant of Ackermann set theory .

Literature (chronological)

  • Zermelo, Ernst: Investigations on the basics 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 Sendet oHG, Walluf 1972, ISBN 3-500-24960-4 .
  • Skolem, Thoralf: About some basic questions of mathematics , 1929, in: selected works in logic, Oslo, 1970, pp. 227-273
  • Zermelo, Ernst: About limit numbers and quantity ranges, in: Fundamenta Mathematicae 16 (1930), pp. 29–47
  • Oberschelp, Arnold: General set theory , Mannheim, Leipzig, Vienna, Zurich, 1994

Individual evidence

  1. Felscher: Naive sets and abstract numbers I, p. 49
  2. Oberschelp, General Set Theory , p. 28
  3. ^ Georg Cantor: Contributions to the foundation of the transfinite set theory . In: Mathematische Annalen 46 (1895), p. 31.
  4. Zermelo: Investigations on the basics of set theory , 1907, in: Mathematische Annalen 65 (1908) , p. 262 (2.)
  5. Zermelo: About limit numbers and quantity ranges, in: Fundamenta Mathematicae 16 (1930), p. 38 possible application.
  6. Jump up ↑ Meschkowsi: Mathematisches Wortswörbuch, Mannheim 1976, p. 279
  7. Oberschelp: Actual classes as primordial elements in set theory , in Mathematische Annalen 157 (1964) , pp. 234–260