Aleph number

The Aleph function, named after the first letter of the Hebrew alphabet and also as written, is an enumeration, more precisely in the theory of cardinal numbers, used in set theory all infinite cardinal numbers.

Definition

The class of infinite cardinal numbers is included by using the axiom of choice in the class of ordinals, each cardinal number is identified with the smallest to the same powerful ordinal. Furthermore, the supremum of a set of cardinal numbers is a cardinal number always again. Therefore, there is exactly one Ordnungsisomorphismus of the class of infinite cardinal numbers. The value of at the location designated by you, that is the -th infinite cardinal.

The Aleph function can be recursively defined as follows:

• = Smallest infinite ordinal, and thus even the smallest infinite cardinal,
• = Smallest cardinal number that is greater than,
• For limit ordinals.

Properties

The smallest infinite cardinal is the cardinality of the countable sets. The successor cardinal number that is the least cardinal number greater than, and so on. The question is equal to the cardinality of the set of real numbers is known as the continuum hypothesis.

Generally, a successor cardinal number, if a successor ordinal, otherwise a limit cardinal number.

Usually refers to the smallest infinite ordinal. This is the same but as an index of the Aleph - function, one prefers to use the ordinal notation. making it the smallest Limes cardinal number and can be written as.

It applies to all of ordinals. It can be shown that there must be fixed points, that is, such ordinal applies. The least fixed point is the limit ( = union ) of the sequence. Weak unattainable cardinal numbers are fixed points of the Aleph function.