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# Difference between revisions of "Aleph-0"

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

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

− | + | the [[cardinality]] of the set of [[natural number]]s. | |

− | + | 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'' | + | a set ''has cardinality aleph-0'') if and only if there is |

− | a set ''has cardinality aleph-0'' | + | a [[bijective function|one-to-one correspondence]] between all elements of the set and all natural numbers. |

− | 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]]; |

− | However, the term "aleph-0" is mainly used in the context of set theory | + | usually the equivalent, but more descriptive term "'''[[countable set|countably infinite]]'''" is used |

− | usually the equivalent, but more descriptive term "'''[[countable set|countably infinite]]'''" is used. | + | (see that article for more details). |

Aleph-0 is the first in the sequence of "small" transfinite numbers, | 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. | the next smallest is aleph-1, followed by aleph-2, and so on. | ||

− | Georg Cantor who first introduced these numbers | + | [[Georg Cantor]], who first introduced these numbers, |

believed aleph-1 to be the cardinality of the set of real 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. | + | (the so-called ''continuum''), but was not able to prove it. |

− | This assumption became known as the [[continuum hypothesis]] | + | This assumption became known as the [[continuum hypothesis]], |

which finally turned out to be independent of the axioms of set theory: | which finally turned out to be independent of the axioms of set theory: | ||

− | First (1938) [[Kurt Gödel]] showed that it cannot be disproved, | + | First (in 1938) [[Kurt Gödel]] showed that it cannot be disproved, |

− | while [[Paul J. Cohen]] much later (1963) | + | while [[Paul J. Cohen]] showed much later (in 1963) that it cannot be proved either. |

+ | |||

<references/> | <references/> |

## Revision as of 09:35, 18 June 2009

In mathematics, **aleph-0** (written and usually pronounced 'aleph null')
^{[1]} is
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
(see that article for more details).

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.