Surreal number

In mathematics, the surreal number system is a totally ordered proper class 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 the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that 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. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. 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 a universal ordered field. In mathematics, the surreal number system is a totally ordered proper class 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 the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that 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. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. 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 a universal ordered field. Research on the go endgame by John Horton Conway led to the original 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. A separate route to defining the surreals began in 1907, when Hans 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 in 1987 he showed that 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. If the surreals are considered as ‘just’ a proper class sized real closed field, Alling’s 1962 paper handles the case of strongly inaccessible cardinals which can naturally be considered as proper classes by cutting off the cumulative hierarchy of the universe one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense. There is an important additional field structure on the surreals that isn’t visible through this lens however, namely the notion of a ‘birthday’ and the corresponding natural description of the surreals as the result of a cut-filling process along their birthdays given by Conway. This additional structure has become fundamental to a modern understanding of the surreal numbers, and Conway is thus given credit for discovering the surreals as we know them today — Alling himself gives Conway full credit in a 1985 paper preceding his book on the subject. 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, 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 { L | R } 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: { L | R } and { L′ | R′ } 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 { L | R } 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