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Aleph

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Aleph' is the first letter of the Hebrew alphabet. Written \aleph.

Usage in set theory

The so-called "variable" is used in mathematics subject of set theory to denote a type of cardinality (set size).

Aleph's come in different relative sizes, each of which is denoted by a subscript. Aleph-sub-zero (\aleph_0), also called Aleph-naught or Aleph-null, is the smallest of the Aleph's. Larger cardinalities are denoted with sequentially higher subscripts.

Though cardinality can be any finite size, the aleph syntax is generally used to denote an infinitude and represent different infinite sizes. Aleph-null is the smallest of "infinities" that lower-level maths generally conceive as infinity (\infty), as the limit of the real values on the number line. However, Aleph's are not indicative of values but rather cardinalities and are reserved for such.

Barring the improper usage of the infinity symbol, and neglecting the fact that the latter is generally used in limits and the former in cardinalities, it is essentially equivalent to say: \aleph_0 = \infty

And it can be proven that:

\aleph_1 = 2^{\aleph_0}

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