Transfinite number: Difference between revisions
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As with finite numbers, there are two ways of thinking of transfinite numbers, as [[ordinality|ordinal]] and [[cardinality|cardinal]] numbers. Unlike the finite ordinals and cardinals, the transfinite ordinals and cardinals define different classes of numbers. |
As with finite numbers, there are two ways of thinking of transfinite numbers, as [[ordinality|ordinal]] and [[cardinality|cardinal]] numbers. Unlike the finite ordinals and cardinals, the transfinite ordinals and cardinals define different classes of numbers. |
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* The lowest transfinite [[ordinal number]] is [[ |
* The lowest transfinite [[ordinal number]] is [[infinity|ω]]. |
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* The first transfinite [[cardinal number]] is [[aleph-null]], <math>\aleph_0</math>, the [[cardinality]] of the [[infinite set]] of the [[integer]]s. The next higher [[cardinal number]] is [[aleph-one]], <math>\aleph_1</math>. |
* The first transfinite [[cardinal number]] is [[aleph-null]], <math>\aleph_0</math>, the [[cardinality]] of the [[infinite set]] of the [[integer]]s. The next higher [[cardinal number]] is [[aleph-one]], <math>\aleph_1</math>. |
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Revision as of 03:47, 1 April 2006
Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary workers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.
As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers. Unlike the finite ordinals and cardinals, the transfinite ordinals and cardinals define different classes of numbers.
- The lowest transfinite ordinal number is ω.
- The first transfinite cardinal number is aleph-null, , the cardinality of the infinite set of the integers. The next higher cardinal number is aleph-one, .
The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers.
Claim of ancient precedent
An early concept of infinite numbers that some scholars consider to be transfinite cardinals is found in India in the works of Jaina mathematicians from the 4th century BC to the 2nd century CE. They classified all numbers into three groups: enumerable, innumerable and infinite; and recognized five different types of infinity: infinite in one direction, infinite in two directions (one dimension), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions).
According to G. G. Joseph and D. P. Agrawal, the highest enumerable number N of the Jains corresponds to the modern concept of aleph-null , the first cardinal transfinite number. The Jains also defined a whole system of infinite numbers, of which the highest enumerable number is the smallest. In the Jaina work on the theory of sets, two basic types of infinite cardinal numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.
References
- Jain, L. C. (1973) "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
- Jain, L. C. (1982) Exact Sciences from Jaina Sources.
- Singh, N. (1987) Jain Theory of Actual Infinity and Transfinite Numbers.
- Joseph, George Gheverghese (2000). The crest of the peacock : the non-European roots of mathematics. Princeton University Press. ISBN 0691006598.
- O'Connor, J. J. and E. F. Robertson (1998) "Georg Ferdinand Ludwig Philipp Cantor", MacTutor History of Mathematics archive.
- Agrawal, D. P. (2000) Ancient Jaina Mathematics: an Introduction, Infinity Foundation.