Aleph-0

In mathematics, aleph-0 (usually pronounced 'aleph null') is the name, and $\aleph _{0}$ the corresponding symbol, used traditionally for the smallest transfinite cardinal number, i.e., for the cardinality of the set of natural numbers. 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, this formulation 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 (and the notation) 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 (1939) Kurt Gödel showed that it cannot be disproved, while J.Paul Cohen much later (1963) showed that it cannot be proved either.