Of fundamental importance crucial, pivotal. (See Wikipedia article on cardinal birds.) (See Wikipedia article on Catholic cardinals.)Ī songbird of the finch family, Cardinalis cardinalis.Īny of various related passerine birds of the family Cardinalidae.
revised (2005), p97Īn official in the Catholic Church, ranking only below the Pope and the patriarchs. The commonest numerals in Latin, as in English, are the "cardinals".and the "ordinals". (See Wikipedia article on Cardinal number.) Referenced on Wolfram|Alpha Cardinal Number Cite this as:įrom MathWorld-A Wolfram Web Resource.Wiktionary Rate this definition: 4.0 / 2 votesĪ number indicating quantity, or the size of a set, e.g., one, two, three. Théorèmes qui équivalent à l'axiome du choix." Fund. Moore, G. H.Īxiom of Choice: Its Origin, Development, and Influence. Of Thought: A History of Set Theory and Its Role in Modern Mathematics. "The Notion of CardinalityĪnd the Continuum Hypothesis." Ch. 6 in Labyrinth Dauben, J. W.Ĭantor: His Mathematics and Philosophy of the Infinite. Oxford,Įngland: Oxford University Press, pp. 83-86, 1996. Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. "Cantor's 'Cardinal Numbers.' " §2.4.3 in What Über unendliche, lineare Punktmannigfaltigkeiten, Arbeiten zur Mengenlehre aus dem Jahren 1872-1884. In terms of a smaller number of smaller cardinals. For any ordinalĪn inaccessible cardinal cannot be expressed Ordered and used this correspondence to define the s ("alephs"). Cantor believed that every set could be well If the axiom of choice is not true, then thereĪre sets that cannot be well ordered. Obviously depends on the axiom of choice, because In this model, the cardinal numbersĪre just the initial ordinals. This leads to the definition of cardinal number for a set as the least ordinalĪre equipollent. That every set can be well ordered and can therefore In fact, the two can be put into one-to-oneĬorrespondence. With the axiom of choice, the cardinalsĬan be enumerated through the ordinals. However, these are two of the more controversial Zermelo-FraenkelĪxioms. It is possible to associate cardinality with a specific set, but the process required either the axiom of foundation or the axiom The problem is that thisĭefinition requires a special axiom to guarantee that cardinals exist.Ī. P. Morse and Dana Scott defined cardinal number by letting be any set, then calling the set of all sets equipollentĪnd of least possible rank (Rubin 1967, p. 270). Is associated with a cardinal number, and two sets and have the same cardinal number iff Tarski (1924) proposed to instead define a cardinal number by stating that every set Unfortunately, the objects producedīy this definition are not sets in the sense of Zermelo-Fraenkel One of the first serious mathematical definitions of cardinal was the one devised by Gottlob Frege and Bertrand Russell, who defined a cardinal number as the set of all sets equipollent The class of real numbers is bigger than the first number class. There is no class bigger than the first number class and smaller than the second.ģ. The second number class is bigger than the first.Ģ. Second number class" (as opposed to the finite ordinals, which he called theġ. that are equipollent to the integers "the Led him to study what would come to be called cardinal numbers. Were not bigger in the sense of equipollence. were bigger than omega in the sense of order, they However, in modern notation, the symbolĬantor, the father of modern set theory, noticed that while the ordinal numbers. Indicated the cardinal number of the set. A double overbar then indicated stripping the order from the set and thus In Georg Cantor's original notation, the symbol for a set annotated with a single overbar indicated stripped of any structure besides order, hence it represented Of a set is also frequently referred to as the "power" of a set (Moore A set has ( aleph-0) members if itĬan be put into a one-to-one correspondence Numbers which are obtainable by counting a given set. Not true for the ordinal numbers.) In fact, theĬardinal numbers are obtained by collecting all ordinal Of counting sets using it gives the same result. In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method In common usage, a cardinal number is a number used in counting (a counting
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