Hartogs' theorem (set theory)

In set theory , Hartogs' theorem (after the German mathematician Fritz Hartogs , 1915) says that for every set A there is at least one well-ordered set B whose cardinality is not restricted by the cardinality of A.

It is remarkable that this statement already applies to the Zermelo-Fraenkel set theory ZF, i.e. it can be proven without using the axiom of choice . Hence, one can use this theorem when examining variants of the axiom of choice. The apparently complicated formulation ("Cardinality of B is not less than or equal to the cardinality of A ") is necessary here because, without an axiom of choice, one cannot show that any two sets are comparable .

Formal statement

${\displaystyle X}$be a set according to the Zermelo-Fraenkel set theory without the axiom of choice. Then there exists a cardinal number (also called Hartogs number of ) such that the set is well ordered and the following applies: ${\displaystyle \alpha }$ ${\displaystyle X}$ ${\displaystyle \alpha }$

• ${\displaystyle \alpha }$is the smallest well-ordered cardinal number that is not less than or equal to the cardinality of (that is, which cannot be mapped into the set injectively .) ${\displaystyle X}$ ${\displaystyle X}$

annotation

In the system ZFC (that ZF + axiom of choice AC) is the set of Hartogs uninteresting because a stronger version than corollary of the well-ordering theorem and the theorem of Cantor follows: For each set X the cardinality is the power set of X is strictly greater than that of X .

literature

• Friedrich Hartogs: About the problem of well-being. Mathematische Annalen Vol. 76, BG Teubner, Leipzig 1915
• Yannis P. Moschovakis: Notes on Set Theory. Springer Verlag, New York 2006, ISBN 0-387-28722-1

Web links

• Mathematische Annalen Vol. 76 at digizeitschriften.de