Aleph-0: Difference between revisions

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''Aleph-0'', usually pronounced 'aleph null', is the [[cardinality]] of the natural numbers. It is the first [[transfinite]] [[Ordinal number|ordinal]]; it represents the "size" of the "smallest" possible infinity. The notion was first introduced by Georg Cantor in his work on the foundations of [[set theory]], and made it possible for mathematicians to reason concretely about the infinite.
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Aleph-0 represents the 'size' of the [[natural numbers]] (0, 1, 2, ...), the [[rational numbers]] (1/2, 2/3, ...), and the [[integers]] (... -1, 0, 1, ...). The size of the [[real numbers]] is in fact strictly bigger, in a sense, than aleph-0; some authors define this set to have cardinality aleph-1. In fact, aleph-0 is the first in an infinite family of infinities, each 'larger' than the last.
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 the traditional notation for 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,
a set ''has cardinality aleph-0'') if and only if there is
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]];
usually the equivalent, but more descriptive term "''[[countable set|countably infinite]]''" is used.


Greek mathematicians first grappled with logical questions about infinity (See [[Zeno]] and [[Archimedes]]) and [[Isaac Newton]] used inadequately defined 'infinitesimals' to develop the [[calculus]]; however over centuries the word ''infinity'' had become so loaded and poorly understood that Cantor himself preferred the term ''transfinite'' to refer to his family of infinities.
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.


[[Category:CZ Live]]
<references/>
[[Category:Mathematics Workgroup]]
[[Category:Stub Articles]]

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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.