Aleph function
The Aleph function , named after the first letter of the Hebrew alphabet and also written as, is an enumeration of all infinite cardinal numbers used in set theory , more precisely in the theory of cardinal numbers.
definition
The class of infinite cardinal numbers is contained in the class of ordinals using the axiom of choice , where each cardinal number is identified with the smallest ordinal number of equal power. Furthermore, the supremum of a set of cardinal numbers is always a cardinal number. Therefore there is exactly one order isomorphism from to the class of the infinite cardinal numbers. The value of at this point is denoted by , that is, the -th infinite cardinal number.
The Aleph function can be defined with transfinite recursion as follows:
- is the smallest infinite ordinal number and therefore also the smallest infinite cardinal number,
- , i.e. the smallest cardinal number that is greater than ,
- for Limes ordinal numbers .
properties
The smallest infinite cardinal number is the cardinality of the countable infinite sets. The successor cardinal number, that is, the smallest cardinal number greater than , is , and so on. The question of whether equals the cardinality of the set of real numbers is known as the continuum hypothesis .
In general , if a successor is ordinal , a successor cardinal number is a Limes cardinal number otherwise .
Usually the smallest denotes an infinite ordinal number. This is the same , but as an index for the Aleph function it is better to use the ordinal number notation. is thus the smallest Limes cardinal number and can be written as.
It always applies to all ordinal numbers . One can show that there must be fixed points , i.e. ordinal numbers for which applies. The smallest fixed point is the limit of the sequence , which is informally represented as . Likewise, weakly unreachable cardinal numbers are fixed points of the Aleph function.
See also
literature
- Georg Cantor : About infinite, linear point manifolds. Work on set theory from the years 1872–1884 (= Teubner Archive for Mathematics. Vol. 2, ISSN 0233-0962 ). Edited and commented by G. Asser . Teubner, Leipzig, 1884.
- Thomas Jech : Set Theory . The Third Millennium Edition, revised and expanded. Springer, Berlin et al. 2003, ISBN 3-540-44085-2 .