# Aleph-0

In mathematics, **aleph-0** (written ℵ_{0} and usually read 'aleph null')
^{[1]}
is the traditional notation for the cardinality of the set of natural numbers.
It is the smallest transfinite cardinal number.
The *cardinality of a set is aleph-0* (or shorter,
a set *has cardinality aleph-0*) if and only if there is
a one-to-one correspondence between all elements of the set and all natural numbers.
However, the term "aleph-0" is mainly used in the context of set theory;
usually the equivalent, but more descriptive term "*countably infinite*" is used.

Aleph-0 is the first in the sequence of "small" transfinite numbers,
the next smallest is aleph-1, followed by aleph-2, and so on.
Georg Cantor, who first introduced these numbers,
believed aleph-1 to be the cardinality of the set of real numbers
(the so-called *continuum*), but was not able to prove it.
This assumption became known as the continuum hypothesis,
which finally turned out to be independent of the axioms of set theory:
First (in 1938) Kurt Gödel showed that it cannot be disproved,
while Paul J. Cohen showed much later (in 1963) that it cannot be proved either.

- ↑
*Aleph*is the first letter of the Hebrew alphabet.

Some content on this page may previously have appeared on Citizendium. |