Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Continuum |
ContinuumNoun1. A continuous nonspatial whole or extent or succession in which no part or portion is distinct of distinguishable from adjacent parts. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "continuum" was first used in popular English literature: sometime before 1686. (references) |
| Domain | Definition |
Aerospace | 1. Something which is continuous, which has no discrete parts, as the continuum of real numbers as opposed to the sequence of discrete integers, as the background continuum of a spectrogram due to thermal radiation.2. = continuous spectrum. (references) |
Biology & Biotechnology | An area over which the vegetation or animal population is of constantly changing composition so that homogeneous, separate communities cannot be distinguished. Source: European Union. (references) |
Electrical Engineering | A compact, connected set. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
See:
- Continuum (mathematics)
- Continuum hypothesis
- Continuum mechanics
- Generalized continuum hypothesis
- Time-space continuum
- Continuum computer game
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Continuum."
(From Wikipedia, the free Encyclopedia)
In mathematics, the word continuum sometimes denotes the real line. Somewhat more generally a continuum is a linearly ordered set that is "densely ordered", i.e., between any two members there is another, and lacks gaps, i.e., every non-empty subset with an upper bound has a least upper bound. By that definition, the long line is a continuum, as are various other sets besides the real line.The "cardinality of the continuum" is the cardinality of the real line. The continuum hypothesis is sometimes stated by saying that now cardinality lies between that of the continuum and that of the natural numbers.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Continuum (mathematics)."
(From Wikipedia, the free Encyclopedia)
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) The continuum hypothesis states the following:
There is no set whose size is strictly between that of the integers and that of the real numbers.Or mathematically speaking, noting that the cardinality for the integers is ("aleph-null") and the cardinality for the real numbers is , the continuum hypothesis says:
The real numbers have also been called the continuum, hence the name. There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis, which is described at the end of this article.
Investigating the continuum hypothesis
Consider the set of all rational numbers. One might naively suppose that there are more rational numbers than integers, and fewer rational numbers than real numbers, thus disproving the continuum hypothesis. However, it turns out that the rational numbers can be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size as the set of integers. (See countable set)
If a set S was found that disproved the continuum hypothesis, it would be impossible to make a one-to-one correspondence between S and the set of integers, because there would always be elements of set S that were "left over". Similarly, it would be impossible to make a one-to-one correspondence between S and the set of real numbers, because there would always be real numbers that were "left over".
Impossibility of proof and disproof
Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. It became the first on David Hilbert's list of important open questions that was presented at the International Mathematical Congress in the year 1900 in Paris.
Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo-Fraenkel set theory axiom system, even if the axiom of choice is adopted. Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is independent of the Zermelo-Fraenkel axiom system and of the axiom of choice. (Both of these results assume that the Zermelo-Fraenkel axioms themselves don't contain a contradiction, something that's widely believed to be true but impossible to prove.)
As such it is not surprising that there should be statements which cannot be proven nor disproven within a given axiom system; in fact the content of Gödel's incompleteness theorem is that such statements always exist if the axiom system is strong enough and without contradictions. The independence of CH was still unsettling however, because it was the first concrete example of an important, interesting question of which it could be proven that it could not be decided either way from the universally accepted basic system of axioms on which mathematics is built.
The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory. As a result of its independence, many substantial conjectures in those fields have subsequently been shown to be independent as well.
It is interesting to note that Gödel believed strongly that CH is false. To him, his independence of proof only showed that the prevalent set of axioms was defective. Gödel was a platonist and therefore had no problems with asserting truth and falsehood of statements independent of their provability. Cohen, however, was a formalist, but even he tended towards rejecting CH. Nowadays, most researchers in the field are either neutral or reject CH. Generally speaking, mathematicians who favour a "rich" and "large" universe of sets are against CH, while those favoring a "neat" and "controllable" universe favor CH. Chris Freiling in 1986 presented an argument against CH: he showed that the negation of CH is equivalent to a statement about probabilities which he calls "intuitively true", but others have disagreed.
To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection S → T. Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Cantor's diagonal argument shows that the integers and the continuum do not have the same cardinality.
The continuum hypothesis states that every subset of the continuum (= the real numbers) which contains the integers either has the same cardinality as the integers or the same cardinality as the continuum.
The generalized continuum hypothesis
The generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S: there are no in-betweens. This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers. GCH is also independent of the Zermelo-Fraenkel set theory axioms and it implies the axiom of choice.
References
- Nancy McGough.: The Continuum Hypothesis, http://www.ii.com/math/ch/
- Cohen, P. J: Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966.
- Gödel, K: The Consistency of the Continuum-Hypothesis. Princeton, NJ: Princeton University Press, 1940.
- Gödel, K.: What is Cantor's Continuum Problem?, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH.
- H. G. Dales and W. H. Woodin: An Introduction to Independence for Analysts. Cambridge (1987).
- Chris Freiling: "Axioms of Symmetry: Throwing Darts at the Real Number Line," Journal of Symbolic Logic, Volume 51 (1986), Issue 1, pp. 190-200.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Continuum hypothesis."
Crosswords: Continuum |
| English words defined with "continuum": brightness ♦ degree ♦ fuzzy logic ♦ history ♦ level ♦ order, order of magnitude ♦ pleasantness, point ♦ stage ♦ time ♦ unpleasantness. (references) |
| Specialty definitions using "continuum": continuous spectrum ♦ discrete spectrum ♦ Environmental gradient ♦ Kirchhoff's second law ♦ launch window, line-reversal pyrometer ♦ rarefied gas dynamics ♦ Space-Time Clustering. (references) |
| Non-English Usage: "Continuum" is also a word in the following languages with English translations in parentheses. Latin (attendant, connected, continuous, hanging together, indivisible, lasting, one who is always around, uninterrupted), Portuguese (continuum). |
| Domain | Usage | |
Screenplays | We should get together and make a continuum! (Friends; writing credit: Jörn O. Jensen; Birger Larsen) My wives exist in different temporal aspects of a four-dimensional space-time continuum. (Goodnight Sweetheart; writing credit: Paul Alexander; Simon Braithwaite) Almost as if it were the junction point for the entire space-time continuum. On the other hand, it could just be an amazing coincidence (Back to the Future Part II; writing credit: Ronnie Cramer) | |
Source: compiled by the editor from various references; see credits. | ||
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Source: compiled by the editor from various references; see credits. | |||
| Thumbnail | Description & Credit | Thumbnail | Description & Credit |
 | Point Udall Millennium Monument. A sundial was erected here, the easternmost point of United States Territory to commemorate the coming of the new millennium . The marker represents "a continuum between all who have come before and all who are yet to come.". Credit: America's Coastlines. |  | Sunrise at the Point Udall Millennium Monument. Here the new day begins for the United States in the Western Hemisphere. The marker represents "a continuum between all who have come before and all who are yet to come.". Credit: America's Coastlines. |
Source: pictures compiled by the editor from various references; see picture credits. | |||
| Subject | Topic | Quote |
Health | Living arrangements representing a continuum of living options that provide assistance services short of institutionalization. (references) | |
Mutations in FBN1, the gene that encodes fibrillin-1, are responsible for MFS and, in a few patients, other disorders in the continuum. (references) | ||
Particular emphasis should be placed on necessity of referral to mental health specialists at various points in the treatment continuum. (references) | ||
Business | There is a continuum from high priority countries, to those which have low latent demand and low accessibility. (references) | |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| Speaker | Term | Phrase(s) |
George Bush | 1989-1993 | I mean that on days like this, we remember that we are all part of a continuum, inescapably connected by the ties that bind. |
Source: compiled by the editor from various references. | ||
| "Continuum" is generally used as a noun (singular) -- approximately 99.76% of the time. "Continuum" is used about 410 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Noun (singular) | 99.76% | 409 | 13,773 |
| Noun (common) | 0.24% | 1 | 339,140 |
| Total | 100.00% | 410 | N/A |
Source: compiled by the editor from several corpora; see credits.
| Hypenated Usage | |
Ending with "continuum": time-continuum. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Language | Translations for "continuum"; alternative meanings/domain in parentheses. | |
Albanian | madhësi e vazhduar. (various references) | |
Chinese | 连续流. (various references) | |
Danish | continuum, kontinuum. (various references) | |
Dutch | continuüm. (various references) | |
Farsi | پیوستگان , پی درپی (Consecutive, Continual, Incessant, Steady), مستمر, تسلسل (Concatenation, Continuity, Progression, Sequence, Suit, Track), زنجیره , رشته مسلسل (Suite). (various references) | |
Finnish | jatkumo. (various references) | |
French | continuum. (various references) | |
German | Kontinuum. (various references) | |
Greek | continuum,συνεχές, συνεχές. (various references) | |
Hungarian | kontinuum (continua), folytonosság (continua, continuance, continuity, perpetuity). (various references) | |
Italian | continuum, continuo (abiding, ceaseless, continual, continuous, hourly, incessant, lasting, non-stop, running, steady). (various references) | |
Japanese Kanji | 連続" (continua). (various references) | |
Japanese Katakana | れ"ぞくたい (continua). (various references) | |
Korean | 연속체. (various references) | |
Pig Latin | ontinuumcay.(various references) | |
Portuguese | continuum, contínuo (attendant, constant, continual, continue, continuous, enduring, eternal, lasting, never-ceasing, ongoing, permanent, perpetual, persistent, progressive, round, running, runny, sequential, solid, straight, sustained, thru, unbroken, unceasing, uninterrupted, unremitting). (various references) | |
Russian | континуум. (various references) | |
Serbo-Croatian | kontinuum, beskonačnost. (various references) | |
Spanish | continuum, zona de transición continua. (various references) | |
Swedish | kontinuum. (various references) | |
Turkish | sürekli (abiding, assiduous, chronic, consistent, consistently, constant, continual, continuous, durable, enduring, everlasting, habitual, hourly, imprescriptible, incessant, invariable, lasting, non-stop, perennial, permanent, perpetual, persistent, running, secular, settled, standing, steady, sustained, unabating, unceasing, unremitting), süreç (course, duration, process), bütün (aggregate, all, all out, all over the, altogether, at all, clear, complement, complete, entire, entirely, every, everything, gross, holo-, integral, omni-, one and only, out and out, overall, pan-, quite, round, sheer, solid, the total, the whole, total, totality, unbroken, undivided, utter, whole, wholly), bölünmemiş şey. (various references) | |
| Source: compiled by the editor from various translation references. | ||
Derivations | |
Words beginning with "continuum": continuums. (additional references) | |
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"Continuum" is suggested in spellcheckers for the following: Cantianum, Canticum, castaneum, contenu, continium, continu, continuam, continuem, continuen, continuim, continum, continumm, continuom, Continuus, continuuum, Londinium, montanum, ponticum, Sentinum. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "c-i-m-n-n-o-t-u-u" | |
-2 letters: unction. | |
-3 letters: conium, muntin, muonic, nuncio. | |
-4 letters: conin, count, cumin, cutin, mount, mucin, muton, niton, notum, onium, ontic, tonic, tunic, uncut, union. | |
-5 letters: cion, coin, coni, conn, icon, into, mint, muni, muon, noun, omit, otic, unci, unco, unit, unto. | |
| Words containing the letters "c-i-m-n-n-o-t-u-u" | |
+1 letter: continuums. | |
+4 letters: multifunction, noncumulative. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Crosswords 3. Usage: Modern 4. Usage: Commercial | 5. Images: Photo Album 6. Quotations: Non-fiction 7. Quotations: Speeches 8. Usage Frequency | 9. Expressions 10. Expressions: Internet 11. Translations: Modern 12. Derivations | 13. Anagrams 14. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
