Aleph-0: Difference between revisions

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(aleph-1 has a standard meaning, and is only equal to the cardinality of the continuum if CH is assumed)
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'''Aleph-0,''' written symbolically <math>\aleph_0</math> and usually pronounced 'aleph null', is the [[cardinality]] of the [[natural number]]s. 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 [[integer]]s (... -1, 0, 1, ...). The size of the [[real number]]s is in fact strictly bigger, in a sense, than aleph-0. 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.


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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.
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