Zermelo 's theorem is a set theory theorem that states that on any set one can introduce such a relation of order that the set is completely ordered . One of the most important theorems in set theory. Named after the German mathematician Ernst Zermelo . Zermelo's theorem is equivalent to the axiom of choice , and therefore to the Zorn lemma .
Content
History
Georg Kantor believed that the statement of this theorem is "a fundamental principle of thought." [1] Indeed, any countable set can be trivially completely ordered, for example, by shifting the order from the set of natural numbers . However, it is difficult for most mathematicians to imagine the complete order already, for example, of the set real numbers. In 1904, announced that he had proved that such an ordering could not exist. A few weeks later, Felix Hausdorff discovered an error in the proof. [2] However, Ernst Zermelo soon published his famous work, [3] in which he proved that any set can be completely ordered. His proof was based on the axiom of choice, first formulated in the same work. The discussion caused by this fact prompted Zermelo subsequently to come to grips with the axiomatization of set theory, which led to the creation of Zermelo – Frenkel axiomatics .
Proof
For the proof, see Assertions equivalent to the axiom of choice .
See also
- A perfectly ordered set
- Axiom of choice
- Lemna Zorn
Literature
- Vereshchagin N. Shen A. The Beginning of Set Theory. - 4th ed. - M.: MCCNMO, 2012 .-- 112 p. - ISBN 978-5-4439-0012-4 .
Notes
- ↑ Georg Cantor (1883), “Ueber unendliche, lineare Punktmannichfaltigkeiten”, Mathematische Annalen 21, pp. 545-591.
- ↑ Plotkin, JM (2005), "Introduction to" The Concept of Power in Set Theory "" , Hausdorff on Ordered Sets , vol. 25, History of Mathematics, American Mathematical Society, p. 23-30, ISBN 9780821890516 , < https://books.google.com/books?id=M_skkA3r-QAC&pg=PA23 >
- ↑ Beweis, dass jede Menge wohlgeordnet werden kann . Mathematische Annalen, 1904.