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# Difference between revisions of "Aleph-0"

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− | In [[mathematics]], '''aleph-0''' (written <math>\aleph_0</math> and usually | + | In [[mathematics]], '''aleph-0''' (written ℵ<sub>0</sub><!--<math>\aleph_0</math>--> and usually read 'aleph null') |

− | <ref> ''Aleph'' is the first letter of the [[Hebrew alphabet]]. </ref> | + | <ref> ''Aleph'' is the first letter of the [[Hebrew alphabet]]. </ref> |

− | the [[cardinality]] of the set of [[natural number]]s. | + | is the traditional notation for the [[cardinality]] of the set of [[natural number]]s. |

It is the smallest transfinite [[cardinal number]]. | It is the smallest transfinite [[cardinal number]]. | ||

The ''cardinality of a set is aleph-0'' (or shorter, | The ''cardinality of a set is aleph-0'' (or shorter, | ||

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a [[bijective function|one-to-one correspondence]] between all elements of the set and all natural numbers. | a [[bijective function|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]]; | However, the term "aleph-0" is mainly used in the context of [[set theory]]; | ||

− | usually the equivalent, but more descriptive term " | + | usually the equivalent, but more descriptive term "''[[countable set|countably infinite]]''" is used. |

Aleph-0 is the first in the sequence of "small" transfinite numbers, | Aleph-0 is the first in the sequence of "small" transfinite numbers, | ||

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First (in 1938) [[Kurt Gödel]] showed that it cannot be disproved, | 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. | while [[Paul J. Cohen]] showed much later (in 1963) that it cannot be proved either. | ||

− | |||

<references/> | <references/> |

## Latest revision as of 18:35, 6 July 2009

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.