Aleph-0: Difference between revisions
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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 03:35, 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.