transfinite number
| Dictionary: transfinite number |
n.
A number that is greater than any finite number.
| 5min Related Video: transfinite number |
| Columbia Encyclopedia: transfinite number |
transfinite number, cardinal or ordinal number designating the magnitude (power) or order of an infinite set; the theory of transfinite numbers was introduced by Georg Cantor in 1874. The cardinal number of the finite set of integers {1, 2, 3, … n} is n, and the cardinal number of any other set of objects that can be put in a one-to-one correspondence with this set is also n; e.g., the cardinal number 5 may be assigned to each of the sets {1, 2, 3, 4, 5}, {2, 4, 6, 8, 10}, {3, 4, 5, 1, 2}, and {a, b, c, d, e}, since each of these sets may be put in a one-to-one correspondence with any of the others. Similarly, the transfinite cardinal number ℵ0 (aleph-null) is assigned to the countably infinite set of all positive integers {1, 2, 3, … n, … }. This set can be put in a one-to-one correspondence with many other infinite sets, e.g., the set of all negative integers {−1, −2, −3, … −n, … }, the set of all even positive integers {2, 4, 6, … 2n, … }, and the set of all squares of positive integers {1, 4, 9, … n2, … }; thus, in contrast to finite sets, two infinite sets, one of which is a subset of the other, can have the same transfinite cardinal number, in this case, ℵ0. It can be proved that all countably infinite sets, among which are the set of all rational numbers and the set of all algebraic numbers, have the cardinal number ℵ0. Since the union of two countably infinite sets is a countably infinite set, ℵ0 + ℵ0 = ℵ0; moreover, ℵ0 × ℵ0 = ℵ0, so that in general, n × ℵ0 = ℵ0 and ℵ0n = ℵ0, where n is any finite number. It can also be shown, however, that the set of all real numbers, designated by c (for "continuum"), is greater than ℵ0; the set of all points on a line and the set of all points on any segment of a line are also designated by the transfinite cardinal number c. An even larger transfinite number is 2c, which designates the set of all subsets of the real numbers, i.e., the set of all {0,1}-valued functions whose domain is the real numbers. Transfinite ordinal numbers are also defined for certain ordered sets, two such being equivalent if there is a one-to-one correspondence between the sets, which preserves the ordering. The transfinite ordinal number of the positive integers is designated by ω.
| Wikipedia: Transfinite number |
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.
Definition
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.
- ω (omega) is defined as the lowest transfinite ordinal number and is the order type of the natural numbers under their usual linear ordering.
- Aleph-null,
, is defined as the first transfinite cardinal number and is the cardinality of the infinite set of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, 
. If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-zero. But in any case, there are no cardinals between aleph-zero and 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. (If Zermelo–Fraenkel set theory (ZFC) is consistent, then neither the continuum hypothesis nor its negation can be proven from ZFC.)
Some authors, for example Suppes, Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set, in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition, the following are all equivalent:
- m is a transfinite cardinal. That is, there is a Dedekind infinite set A such that the cardinality of A is m.
- m + 1 = m.
- ≤ m.
- there is a cardinal n such that + n = m.
See also
|
References
- Levy, Azriel, 2002 (1978) Basic Set Theory. Dover Publications. ISBN 0-486-42079-5
- O'Connor, J. J. and E. F. Robertson (1998) "Georg Ferdinand Ludwig Philipp Cantor," MacTutor History of Mathematics archive.
- Rubin, Jean E., 1967. "Set Theory for the Mathematician". San Francisco: Holden-Day. Grounded in Morse-Kelley set theory.
- Rudy Rucker, 2005 (1982) Infinity and the Mind. Princeton Univ. Press. Primarily an exploration of the philosophical implications of Cantor's paradise.
- Patrick Suppes, 1972 (1960) "Axiomatic Set Theory". Dover. ISBN 0-486-61630-4. Grounded in ZFC.
|
|||||||||||
) · 
)
)
or 
) ·
)
)
)
)
)
)
)
|
|||||||||||||||||||
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
| Best of the Web: transfinite number |
Some good "transfinite number" pages on the web:
 Math mathworld.wolfram.com |
Learn More
 | infinity (in mathematics) |
| number (philosophy) |
| Transfinite arithmetic |
 | Where can you get her number? Read answer... |
| Can you have he number? Read answer... |
| What number are you on? Read answer... |
Help us answer these
 | What is it's number? |
| What i your number? |
| What is the answer to number? |
Post a question - any question - to the WikiAnswers community:
Copyrights:
 |
 | Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more |
 |
| Columbia Encyclopedia. The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/. Read more |
 |
| Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Transfinite number". Read more |
Related answers
- Is graham number beyond all finite transfinite and transinfinite numbers?
- What year did Cantor create transfinite numbers?
- By the numbers?
ADVERTISEMENT
Answer these
- Does a number?
- How do you get a number?
- What are the numbers?
- What is your number's?
- Do you have a number for it?
- How do you get your number?
Mentioned in
- infinity (in mathematics)
- number (philosophy)
- Transfinite arithmetic
- Elwick
