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Difference between revisions of "Aleph-0"

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In [[mathematics]], '''aleph-0''' (usually pronounced 'aleph null')
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In [[mathematics]], '''aleph-0''' (written <math>\aleph_0</math> and usually pronounced 'aleph null')
<ref> 'aleph' is the first letter of the Hebrew alphabet </ref> is the name,
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<ref> ''Aleph'' is the first letter of the [[Hebrew alphabet]]. </ref> is  
and <math>\aleph_0</math> the corresponding symbol, used traditionally
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the [[cardinality]] of the set of [[natural number]]s.  
for the smallest transfinite [[cardinal number]],
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It is the smallest transfinite [[cardinal number]].
i.e., for the [[cardinality]] of the set of natural numbers.
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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 set ''has cardinality aleph-0'') if and only if there is  
a set ''has cardinality aleph-0'', if and only if there is  
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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.
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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,
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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.
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(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 (and the notation)
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[[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.
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(the so-called ''continuum''), but was not able to prove it.
This assumption became known as the [[continuum hypothesis]]
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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,
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First (in 1938) [[Kurt Gödel]] showed that it cannot be disproved,
while [[Paul J. Cohen]] much later (1963) showed that it cannot be proved either.
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while [[Paul J. Cohen]] showed much later (in 1963) that it cannot be proved either.
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<references/>
 
<references/>

Revision as of 09:35, 18 June 2009

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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.


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