alephnull

Aleph-null, also written aleph0 or aleph-zero and denoted by the symbol ℵ0, is the smallest infinite cardinal number. It is the cardinality of the set of natural numbers N, and therefore the size of any countably infinite set (for example Z, the integers, and Q, the rationals).

In set theory, aleph-null is the first in the sequence of aleph numbers and is an initial cardinal. Its corresponding initial ordinal is ω, the first infinite ordinal; while ω has cardinality aleph-null.

Cantor's diagonal argument shows that the set of real numbers R has cardinality 2^ℵ0, which is strictly greater than aleph-null; thus the real numbers form an uncountable set. The continuum c is often used for 2^aleph0 and is the cardinality of the set of real numbers. The continuum hypothesis asks whether 2^aleph0 equals aleph1, and this is independent of ZFC.

Some standard closure properties: the union of countably many countable sets is countable; the Cartesian product of two countable sets is countable; there exist countable infinite sets besides N, all having cardinal aleph-null. In mathematics, aleph-null serves as a baseline for measuring the size of infinity.

Thus aleph-null is fundamental in set theory as the size of the most basic infinite set and as a reference point for comparing infinite sizes.