Aleph-0: Difference between revisions
imported>Meni Rosenfeld (aleph-1 has a standard meaning, and is only equal to the cardinality of the continuum if CH is assumed) |
imported>Ragnar Schroder (Added sections and some links.) |
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'''Aleph-0 | '''Aleph-0''' is a formal mathematical term describing in a technical sense the "size" of the set of all integers. | ||
==Introduction== | |||
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. | |||
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. | 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. | ||
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. | 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. | ||
== See also == | |||
*[[Countably infinite]] | |||
*[[Transfinite cardinals]] | |||
*[[Continuum hypothesis]] | |||
==Related topics== | |||
*[[Hilbert's hotel]] | |||
*[[Galileo's paradox]] | |||
*[[Georg Cantor]] | |||
== References== | |||
== External links == | |||
#[http://mathworld.wolfram.com/Aleph-1.html] | |||
[[Category:CZ Live]] | [[Category:CZ Live]] | ||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
[[Category:Stub Articles]] | [[Category:Stub Articles]] | ||
Revision as of 07:38, 15 November 2007
Aleph-0 is a formal mathematical term describing in a technical sense the "size" of the set of all integers.
Introduction
Aleph-0, written symbolically and usually pronounced 'aleph null', is the cardinality of the natural numbers. It is the first transfinite 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.
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. In fact, aleph-0 is the first in an infinite family of infinities, each 'larger' than the last.
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.
See also
Related topics