What is "aleph number"

n. (context set theory English) Any of a sequence of numbers used to represent the cardinality of infinite sets, denoted by the Hebrew letter aleph.

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 (ℵ) (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 ℵ (read aleph-naught or aleph-zero; the German term aleph-null is also sometimes used), the next larger cardinality is aleph-one ℵ, then ℵ and so on. Continuing in this manner, it is possible to define a cardinal number ℵ for every ordinal number α, as described below.

The concept and notation are due to Georg Cantor, Miller quotes : "His new numbers deserved something unique. ... Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet...the aleph could be taken to represent new beginnings..."

who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that " diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.