Kurepa's theorem

The theorem of Kurepa ( English Theorem of Kurepa ) is a mathematical theorem from the branch of set theory . It goes back to the Yugoslav mathematician Đuro Kurepa .

The sentence contains a logically equivalent formulation of the axiom of choice in the language of order theory .

Formulation of the sentence

Kurepa's theorem can be formulated as follows:

The axiom of choice is logically equivalent to the condition that each of the following two principles (   ) and has     validity: ${\displaystyle {V}}$ ${\displaystyle ({\overline {K}})}$

  ${\displaystyle (V)}$    : There   is a linear order on every set   . ${\displaystyle X}$   ${\displaystyle \leq }$

  ${\displaystyle ({\overline {K}})}$  : Each anti chain of each partially ordered set is in a relative maximum antichain included . ${\displaystyle (X,\leq )}$ ${\displaystyle \subseteq }$

In a formulaic brief description, the sentence can also be stated as follows:

Axiom of choice   ${\displaystyle \iff }$     ${\displaystyle ({V})\land ({\overline {K}})}$

literature

Original work

• G. Kurepa: On the Axiom of Choice . In: Math. Ann. tape 126 , 1953, pp. 381-384 ( MR0058686 ).

Monographs

• Egbert Harzheim : Ordered Sets (=  Advances in Mathematics . Volume 7 ). Springer Verlag, New York 2005, ISBN 0-387-24219-8 , pp. 206 ff . ( MR2127991 ).
• Wacław Sierpiński : Cardinal and Ordinal Numbers (=  Monograph Matematyczne . Volume 34 ). 2nd Edition. Panstwowe Wydawnictwo Naukowe, Warsaw 1965 ( MR0194339 ).

Individual evidence

1. ↑ ^{a b} Harzheim: p. 52.
2. ↑ ^{a b} Sierpiński, p. 428
3. ↑ Often also called under the name Đuro Kurepa or (mostly in English-speaking countries) under Djuro Kurepa; Cyrillic Ђуро Курепа (* August 16, 1907, † November 2, 1993) - Dura Kurepa. history.mcs.st-andrews.ac.uk
4. ^ Kurepa: On the Axiom of Choice . In: Math. Ann . tape 126 , 1953, pp. 381 .