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In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by labelling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange infinite collections of objects in order. Ordinals are distinct from cardinal numbers, which are useful for saying how many objects are in a collection. Although the distinction between ordinals and cardinals is not always apparent in finite sets (one can go from one to the other just by counting labels), different infinite ordinals can describe the same cardinal (see Hilbert's grand hotel). Ordinals were introduced by Georg Cantor in 1883〔Thorough introductions are given by Levy (1979) and Jech (2003).〕 to accommodate infinite sequences and to classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.〔. See the footnote on p. 12.〕 Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is ω, which is identified with the cardinal number . However, in the transfinite case, beyond ω, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely itself, there are uncountably many countably infinite ordinals, namely :ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε0, …. Here addition and multiplication are not commutative: in particular 1 + ω is ω rather than ω + 1 and likewise, 2·ω is ω rather than ω·2. The set of all countable ordinals constitutes the first uncountable ordinal ω1, which is identified with the cardinal (next cardinal after ). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The ''cardinality of an ordinal'' defines a many to one association from ordinals to cardinals. In general, each ordinal α is the ''order type of the set of ordinals'' strictly less than the ordinal α itself. This property permits every ordinal to be represented as the set of all ordinals less than it. Ordinals may be categorized as: zero, successor ordinals, and limit ordinals (of various cofinalities). Given a ''class of ordinals'', one can identify the α-th member of that class, i.e. one can index (count) them. Larger and larger ordinals can be defined, but they become more and more difficult to describe. == Ordinals extend the natural numbers == A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the ''size'' of a set, or to describe the ''position'' of an element in a sequence. When restricted to finite sets these two concepts coincide, there is only one way to put a finite set into a linear sequence, up to isomorphism. When dealing with infinite sets one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which is generalized by the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below. Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way) in which there is no infinite ''decreasing'' sequence (however, there may be infinite increasing sequences); equivalently, every non-empty subset of the set has a least element. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on") and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the ''order type'' of the set. Any ordinal is defined by the set of ordinals that precede it: in fact, the most common definition of ordinals ''identifies'' each ordinal ''as'' the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less than it, i.e., the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it is generally identified as the set . Conversely, any set (''S'') of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is also in S—is (or can be identified with) an ordinal. There are infinite ordinals as well: the smallest infinite ordinal is ω, which is the order type of the natural numbers (finite ordinals) and that can even be identified with the ''set'' of natural numbers (indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed it can be identified with the ordinal associated with it, which is exactly how ω is defined). Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After ''all'' natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals formed in this way (the ω·''m''+''n'', where ''m'' and ''n'' are natural numbers) must itself have an ordinal associated with it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωω², and much later on ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely far ("indefinitely far" is exactly what ordinals are good at: basically every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ordinal number」の詳細全文を読む スポンサード リンク
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