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. ${\displaystyle \aleph }$

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. ${\displaystyle \mathbf {On} }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \aleph }$ ${\displaystyle \mathbf {On} }$ ${\displaystyle \aleph }$ ${\displaystyle \alpha }$ ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \alpha }$

The Aleph function can be defined with transfinite recursion as follows:

• ${\displaystyle \aleph _{0}=|\omega |}$ is the smallest infinite ordinal number and therefore also the smallest infinite cardinal number,
• ${\displaystyle \aleph _{\alpha +1}=\min _{\kappa >\aleph _{\alpha }}\kappa }$, i.e. the smallest cardinal number that is greater than , ${\displaystyle \aleph _{\alpha }}$
• ${\displaystyle \aleph _{\lambda }=\sup _{\alpha <\lambda }\aleph _{\alpha }}$for Limes ordinal numbers . ${\displaystyle \lambda }$

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 . ${\displaystyle \aleph _{0}}$ ${\displaystyle \aleph _{0}}$ ${\displaystyle \aleph _{1}}$ ${\displaystyle \aleph _{1}}$

In general , if a successor is ordinal , a successor cardinal number is a Limes cardinal number otherwise . ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \alpha }$

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. ${\displaystyle \omega }$ ${\displaystyle \aleph _{0}}$ ${\displaystyle \aleph _{\omega }}$ ${\displaystyle \sup _{n<\omega }\aleph _{n}}$

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. ${\displaystyle \alpha \leq \aleph _{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha =\aleph _{\alpha }}$ ${\displaystyle \aleph _{0},\aleph _{\aleph _{0}},\aleph _{\aleph _{\aleph _{0}}},\ldots }$ ${\displaystyle \aleph _{\aleph _{{}_{\ddots }}}}$

See also

• Starting number
• Beth function
• Gimel function

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 .