Cardinal Numbers
Cardinal numbers describe the size of a setor collection, for instance how many dollars or how many days there are. The word cardinal comes from the Latin cardinalis for "most basic" or "most important," which underlines the significance people give to the idea of "how many."
When people count, they are actually using a combination of cardinal and ordinal numbers, two key mathematical concepts. Ordinal numbers also tell "how many" but in the context of "in what order"--for example, which day of the month it is. The Latin ordinalis means "order," as in a serial arrangement. Thus, when people count a collection of items, they are using cardinal numbers to tell how many, but the counting system itself is ordinal, because the numbers used for counting must be used in a specific sequence to be meaningful. It is possible for a set to contain only the number 0. Zero tells "how many" and is thus a cardinal number.
When people are trying to compare the size of two sets or collections, usually they will count "how many" in each collection and then compare the cardinal numbers. Cardinal numbers are an important part of set theory, which is a method of studying and explaining the basic properties of arithmetic. In set theory, a cardinal number is the number obtained by counting a set. If counting the set always yields the same result, then that result is a cardinal number. One could say that this is simply the theoretician's way of explaining "how many."
Although many math courses begin with an overview of cardinal numbers and set theory, theoreticians think set theory looks deceptively simple on the surface. They point to the impossibility of defining a cardinal number without first establishing many elaborate preconditions. Gottlob Frege(1848-1925) and Bertrand Russell(1872-1970) were among the first mathematicians to attempt such a definition. They said a cardinal number A consisted of all sets equipollent (of equal signification) to A. Again, one could boil this down to a consideration of "how many." Mathematicians disputed the definition of Frege and Russell on the grounds that it does not consider sets in the traditional way. Mathematics texts offer lengthy definitions and conditions for cardinal numbers, with variations such as finite and infinite cardinals, Dedekind-finite cardinals, successor cardinals, regular cardinals, and inaccessible cardinals.
The terms cardinal and ordinal are so closely linked that it is difficult to explain one without drawing on the other. In every society, in every language, the challenge is still unmet[emsp3 ]no one has found a way to explain cardinal and ordinal as independent concepts. A mathematician would say that all of the nonnegative whole numbers (0, 1, 2, 3, 4...) are simultaneously cardinal and ordinal.
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