Aleph number

In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They are named after the symbol used to denote them, the Hebrew letter aleph (${\displaystyle \aleph }$) (though in older mathematics books the letter aleph is often printed upside down by accident, partly because a Monotype matrix for aleph was mistakenly constructed the wrong way up ).

The cardinality of the natural numbers is ${\displaystyle \aleph _{0}}$ (read aleph-naught or aleph-zero; the German term aleph-null is also sometimes used), the next larger cardinality is aleph-one ${\displaystyle \aleph _{1}}$, then ${\displaystyle \aleph _{2}}$ and so on. Continuing in this manner, it is possible to define a cardinal number ${\displaystyle \aleph _{\alpha }}$ for every ordinal number α, as described below.

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