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In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. (Strictly speaking, the surreals are not a set, but a proper class.〔In the original formulation using von Neumann–Bernays–Gödel set theory, the surreals form a proper class, rather than a set, so the term field is not precisely correct; where this distinction is important, some authors use Field or FIELD to refer to a proper class that has the arithmetic properties of a field. One can obtain a true field by limiting the construction to a Grothendieck universe, yielding a set with the cardinality of some strongly inaccessible cardinal, or by using a form of set theory in which constructions by transfinite recursion stop at some countable ordinal such as epsilon nought.〕) If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are the largest ordered field. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. In 1907 Hahn introduced Hahn series as a generalization of formal power series, and Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962 Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α, and taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers. Research on the go endgame by John Horton Conway led to another definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book ''Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness''. In his book, which takes the form of a dialogue, Knuth coined the term ''surreal numbers'' for what Conway had called simply ''numbers''. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book ''On Numbers and Games''. ==Overview== In the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers ''a'' and ''b'' either ''a'' ≤ ''b'' or ''b'' ≤ ''a''. (Both may hold, in which case ''a'' and ''b'' are equivalent and denote the same number.) Numbers are formed by pairing subsets of numbers already constructed: given subsets ''L'' and ''R'' of numbers such that all the members of ''L'' are strictly less than all the members of ''R'', then the pair represents a number intermediate in value between all the members of ''L'' and all the members of ''R''. Different subsets may end up defining the same number: and may define the same number even if ''L'' ≠ ''L′'' and ''R'' ≠ ''R′''. (A similar phenomenon occurs when rational numbers are defined as quotients of integers: 1/2 and 2/4 are different representations of the same rational number.) So strictly speaking, the surreal numbers are equivalence classes of representations of form that designate the same number. In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: . This representation, where ''L'' and ''R'' are both empty, is called 0. Subsequent stages yield forms like: : = 1 : = 2 : = 3 and : = −1 : = −2 : = −3 The integers are thus contained within the surreal numbers. (The above identities are definitions, in the sense that the right-hand side is a name for the left-hand side. That the names are actually appropriate will be evident when the arithmetic operations on surreal numbers are defined, as in the section below). Similarly, representations arise like: : = 1/2 : = 1/4 : = 3/4 so that the dyadic rationals (rational numbers whose denominators are powers of 2) are contained within the surreal numbers. After an infinite number of stages, infinite subsets become available, so that any real number ''a'' can be represented by , where ''La'' is the set of all dyadic rationals less than ''a'' and ''Ra'' is the set of all dyadic rationals greater than ''a'' (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals. But there are also representations like : = ω : = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth. 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「Surreal number」の詳細全文を読む スポンサード リンク
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