Aleph-0: Difference between revisions

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'''Aleph-0''' – notated <math>\aleph_0</math> – is a formal mathematical term describing in a technical sense the "size" of the set of all integers.
In [[mathematics]], '''aleph-0''' (usually pronounced 'aleph null') is the name,
and <math>\aleph_0</math> the corresponding symbol, used traditionally
for the smallest transfinite [[cardinal number]],
i.e., for the [[cardinality]] of the set of natural numbers.
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, this formulation is mainly used in the context of set theory,
usually the equivalent, but more descriptive term '''[[countable set|countably infinite]]''' is used.


==Introduction==
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.
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]] [[cardinal number]]; 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.
Georg Cantor who first introduced these numbers (and the notation)
 
believed aleph-1 to be the cardinality of the set of real numbers
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.
(the so called ''continuum''), but was not able to prove it.
 
This assumption became known as the [[continuum hypothesis]]
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.
which finally turned out to be independent of the axioms of set theory:
First (1939) [[Kurt Gödel]] showed that it cannot be disproved,
while J.[[Paul Cohen]] much later (1963) showed that it cannot be proved either.

Revision as of 17:48, 11 June 2009

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In mathematics, aleph-0 (usually pronounced 'aleph null') is the name, and the corresponding symbol, used traditionally for the smallest transfinite cardinal number, i.e., for the cardinality of the set of natural numbers. 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, this formulation 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 (and the notation) 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 (1939) Kurt Gödel showed that it cannot be disproved, while J.Paul Cohen much later (1963) showed that it cannot be proved either.

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