Hartogs number

In set theory, the set of Hartogs ( after the German mathematician Fritz Hartogs, 1915) states that there are lots of A is at least a well-ordered set B whose cardinality is not limited by the cardinality of A.

It is noteworthy that this statement is already in the Zermelo -Fraenkel set theory ZF, ie can be proved without using the axiom of choice. Therefore, one can use this set, if one examines variants of the axiom of choice. The seemingly complicated wording (" cardinality of B is not less than or equal to the cardinality of A") is necessary here, because you can not show without axiom of choice that any two quantities are comparable.

Formal statement

Is 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 designated ) such that the quantity is well-ordered and:

• Is the smallest well-ordered cardinal number which is not less than or equal to the cardinality of ( ie: . injective which can not be mapped into the crowd )

Note

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