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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 pronounced 'aleph null')
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In [[mathematics]], '''aleph-0''' (written &alefsym;<sub>0</sub><!--<math>\aleph_0</math>--> and usually read 'aleph null')
<ref> ''Aleph'' is the first letter of the [[Hebrew alphabet]]. </ref> is
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<ref> ''Aleph'' is the first letter of the [[Hebrew alphabet]]. </ref>
the [[cardinality]] of the set of [[natural number]]s.  
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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 "'''[[countable set|countably infinite]]'''" is used.
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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

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In mathematics, aleph-0 (written ℵ0 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.

  1. Aleph is the first letter of the Hebrew alphabet.