Aleph-0: Difference between revisions
imported>Larry Sanger (Added symbol; we may or may not want to translate the other uses) |
imported>Meni Rosenfeld (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. | '''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 | 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. | ||
Revision as of 03:42, 9 November 2007
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.