# 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}}}}$