In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of 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.
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 (including 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.
History of the concept 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 Felix Hausdorff introduced certain ordered sets called -sets for ordinals and asked if it was possible to find a compatible ordered group or field structure. In 1962, Norman Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals and, in 1987, he showed that taking 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 is not 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.
Description
Notation In the context of surreal numbers, an ordered pair of sets of surreal numbers, and , which is written as in many other mathematical contexts, is instead written including the extra space adjacent to each brace. When or is explicitly described by its elements, the pair of braces that encloses the set of surreal elements is often omitted. When or is empty, it is often simply omitted. For example, instead of , which is common notation in other contexts, we typically write , where , , and are surreal numbers.
Outline of construction In the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers and , or . (Both may hold, in which case and are equivalent and denote the same number.) Each number is formed from an ordered pair of subsets of numbers already constructed: given subsets and of numbers such that all the members of are strictly less than all the members of , then the pair represents a number intermediate in value between all the members of and all the members of . According to Conway, intermediate values must be governed by his rule of simplicity. That is, numbers born on subsequent birthdays must be the simplest between the new and the prior day. For example, on day 1, -1 and 1 are born. On day 2, the simplest number between 1 and 0 is 1/2; between 0 and -1 is -1/2. Different subsets may end up defining the same number: and may define the same number even if and . (A similar phenomenon occurs when rational numbers are defined as quotients of integers: and are different representations of the same rational number.) Each surreal number is an equivalence class of representations of the form that designate the same number, noting that each equivalence class is a proper class rather than a set.
In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: . This representation, where and are both empty, is called 0. Subsequent stages yield forms like
and
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 such as
arise, 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 can be represented by where is the set of all dyadic rationals less than and is the set of all dyadic rationals greater than (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.
There are also representations like
where is a transfinite number greater than all integers and 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 or and so forth.
Construction
Surreal numbers are constructed inductively as equivalence classes of pairs of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set. The construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.
Forms A form is a pair of sets of surreal numbers, called its left set and its right set. A form with left set and right set is written . When and are given as lists of elements, the braces around them are omitted.
Either or both of the left and right set of a form may be the empty set. The form with both left and right set empty is also written .
Numeric forms and their equivalence classes
Construction rule
The numeric forms are placed in equivalence classes; each such equivalence class is a surreal number. The elements of the left and right sets of a form are drawn from the universe of the surreal numbers (not of forms, but of their equivalence classes).
Equivalence rule
An ordering relationship must be antisymmetric, i.e., it must have the property that (i. e., and are both true) only when and are the same object. This is not the case for surreal number forms, but is true by construction for surreal numbers (equivalence classes).
The equivalence class containing is labeled 0; in other words, is a form of the surreal number 0.
Order
The recursive definition of surreal numbers is completed by defining comparison:
Given numeric forms and , if and only if both: There is no such that . That is, every element in the left part of is strictly smaller than . There is no such that . That is, every element in the right part of is strictly larger than . Surreal numbers can be compared to each other (or to numeric forms) by choosing a numeric form from its equivalence class to represent each surreal number.
Induction
This group of definitions is recursive, and requires some form of mathematical induction to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable via finite induction are the dyadic fractions; a wider universe is reachable given some form of transfinite induction.
Induction rule There is a generation , in which 0 consists of the single form . Given any ordinal number , the generation is the set of all surreal numbers that are generated by the construction rule from subsets of .
The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no with , the expression is the empty set; the only subset of the empty set is the empty set, and therefore consists of a single surreal form lying in a single equivalence class 0.
For every finite ordinal number , is well-ordered by the ordering induced by the comparison rule on the surreal numbers.
The first iteration of the induction rule produces the three numeric forms (the form is non-numeric because ). The equivalence class containing is labeled 1 and the equivalence class containing is labeled −1. These three labels have a special significance in the axioms that define a ring; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels.
For every , since every valid form in is also a valid form in , all of the numbers in also appear in (as supersets of their representation in ). (The set union expression appears in our construction rule, rather than the simpler form , so that the definition also makes sense when is a limit ordinal.) Numbers in that are a superset of some number in are said to have been inherited from generation . The smallest value of for which a given surreal number appears in is called its birthday. For example, the birthday of 0 is 0, and the birthday of −1 is 1.
A second iteration of the construction rule yields the following ordering of equivalence classes:
Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow: contains four new surreal numbers. Two contain extremal forms: contains all numbers from previous generations in its right set, and contains all numbers from previous generations in its left set. The others have a form that partitions all numbers from previous generations into two non-empty sets. Every surreal number that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers other than from previous generations into a left set (all numbers less than ) and a right set (all numbers greater than ). The equivalence class of a number depends on only the maximal element of its left set and the minimal element of the right set.
The informal interpretations of and are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of and are "the simplest number between 0 and 1" and "the simplest number between −1 and 0" respectively; their equivalence classes are labeled and −. These labels will also be justified by the rules for surreal addition and multiplication below.