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 ηα-sets for ordinals α 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 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 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, L and R, which is written as (L, R) in many other mathematical contexts, is instead written { L | R } including the extra space adjacent to each brace. When L or R is explicitly described by its elements, the pair of braces that encloses the set of surreal elements is often omitted. When L or R is empty, it is often simply omitted. For example, instead of ({0, 1, 2}, {}), which is common notation in other contexts, we typically write { 0, 1, 2 | }, where 0, 1, and 2 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 a and b, a ≤ b or b ≤ a. (Both may hold, in which case a and b are equivalent and denote the same number.) Each number is formed from an ordered pair of 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. 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: { 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.) Each surreal number is an equivalence class of representations of the form { L | R } 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 L and R 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 a can be represented by { La | Ra }, 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. There are also representations like
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.
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 L and right set R is written { L | R }. When L and R 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 x = y (i. e., x ≤ y and y ≤ x are both true) only when x and y 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 x = { XL | XR } and y = { YL | YR }, x ≤ y if and only if both:
There is no xL ∈ XL such that y ≤ xL. That is, every element in the left part of x is strictly smaller than y. There is no yR ∈ YR such that yR ≤ x. That is, every element in the right part of y is strictly larger than x. 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 S0 = { 0 }, in which 0 consists of the single form { | }. Given any ordinal number n, the generation Sn is the set of all surreal numbers that are generated by the construction rule from subsets of ⋃ i < n S i {\textstyle \bigcup _{i<n}S_{i}} . 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 Si with i < 0, the expression ⋃ i < 0 S i {\textstyle \bigcup _{i<0}S_{i}} is the empty set; the only subset of the empty set is the empty set, and therefore S0 consists of a single surreal form { | } lying in a single equivalence class 0. For every finite ordinal number n, Sn 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 { | 0 } < { | } < { 0 | } (the form { 0 | 0 } is non-numeric because 0 ≤ 0). The equivalence class containing { 0 | } is labeled 1 and the equivalence class containing { | 0 } 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 i < n, since every valid form in Si is also a valid form in Sn, all of the numbers in Si also appear in Sn (as supersets of their representation in Si). (The set union expression appears in our construction rule, rather than the simpler form Sn−1, so that the definition also makes sense when n is a limit ordinal.) Numbers in Sn that are a superset of some number in Si are said to have been inherited from generation i. The smallest value of α for which a given surreal number appears in Sα 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:
S2 contains four new surreal numbers. Two contain extremal forms: { | −1, 0, 1 } contains all numbers from previous generations in its right set, and { −1, 0, 1 | } 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 x 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 x from previous generations into a left set (all numbers less than x) and a right set (all numbers greater than x). 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 { 1 | } and { | −1 } 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 { 0 | 1 } and { −1 | 0 } are "the simplest number between 0 and 1" and "the simplest number between −1 and 0" respectively; their equivalence classes are labeled 1/2 and −1/2. These labels will also be justified by the rules for surreal addition and multiplication below. The equivalence classes at each stage n of induction may be characterized by their n-complete forms (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains every number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels:
The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number { 1, 2 | 5, 8 } is therefore equivalent to { 2 | 5 }; one can establish that these are forms of 3 by using the birthday property, which is a consequence of the rules above.
Birthday property A form x = { L | R } occurring in generation n represents a number inherited from an earlier generation i < n if and only if there is some number in Si that is greater than all elements of L and less than all elements of the R. (In other words, if L and R are already separated by a number created at an earlier stage, then x does not represent a new number but one already constructed.) If x represents a number from any generation earlier than n, there is a least such generation i, and between L and R lies exactly one number c that has this least i as its birthday. x is a form of this c. In other words, it lies in the equivalence class in Sn that is a superset of the representation of c in generation i.
Arithmetic The addition, negation (additive inverse), and multiplication of surreal number forms x = { XL | XR } and y = { YL | YR } are defined by three recursive formulas.
Negation Negation of a given number x = { XL | XR } is defined by
− x = − { X L ∣ X R } = { − X R ∣ − X L } , {\displaystyle -x=-\{X_{L}\mid X_{R}\}=\{-X_{R}\mid -X_{L}\},}
where the negation of a set S of numbers is given by the set of the negated elements of S:
− S = { − s : s ∈ S } . {\displaystyle -S=\{-s:s\in S\}.}
This formula involves the negation of the surreal numbers appearing in the left and right sets of x, which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form. This makes sense only if the result is the same, irrespective of the choice of form of the operand. This can be proved inductively using the fact that the numbers occurring in XL and XR are drawn from generations earlier than that in which the form x first occurs, and observing the special case:
− 0 = − {
∣
} = {
∣
} = 0. {\displaystyle -0=-\{{}\mid {}\}=\{{}\mid {}\}=0.}
Addition The definition of addition is also a recursive formula:
x + y = { X L ∣ X R } + { Y L ∣ Y R } = { X L + y , x + Y L ∣ X R + y , x + Y R } , {\displaystyle x+y=\{X_{L}\mid X_{R}\}+\{Y_{L}\mid Y_{R}\}=\{X_{L}+y,x+Y_{L}\mid X_{R}+y,x+Y_{R}\},}
where
X + y = { x ′ + y : x ′ ∈ X } , x + Y = { x + y ′ : y ′ ∈ Y } {\displaystyle X+y=\{x'+y:x'\in X\},\quad x+Y=\{x+y':y'\in Y\}}
This formula involves sums of one of the original operands and a surreal number drawn from the left or right set of the other. It can be proved inductively with the special cases:
0 + 0 = {
∣
} + {
∣
} = {
∣
} = 0 {\displaystyle 0+0=\{{}\mid {}\}+\{{}\mid {}\}=\{{}\mid {}\}=0}
x + 0 = x + {
∣
} = { X L + 0 ∣ X R + 0 } = { X L ∣ X R } = x {\displaystyle x+0=x+\{{}\mid {}\}=\{X_{L}+0\mid X_{R}+0\}=\{X_{L}\mid X_{R}\}=x}
0 + y = {
∣
} + y = { 0 + Y L ∣ 0 + Y R } = { Y L ∣ Y R } = y {\displaystyle 0+y=\{{}\mid {}\}+y=\{0+Y_{L}\mid 0+Y_{R}\}=\{Y_{L}\mid Y_{R}\}=y}
For example:
which by the birthday property is a form of 1. This justifies the label used in the previous section.
Subtraction Subtraction is defined with addition and negation:
x − y = { X L ∣ X R } + { − Y R ∣ − Y L } = { X L − y , x − Y R ∣ X R − y , x − Y L } . {\displaystyle x-y=\{X_{L}\mid X_{R}\}+\{-Y_{R}\mid -Y_{L}\}=\{X_{L}-y,x-Y_{R}\mid X_{R}-y,x-Y_{L}\}\,.}
Multiplication Multiplication can be defined recursively as well, beginning from the special cases involving 0, the multiplicative identity 1, and its additive inverse −1:
x y = { X L ∣ X R } { Y L ∣ Y R } = { X L y + x Y L − X L Y L , X R y + x Y R − X R Y R ∣ X L y + x Y R − X L Y R , x Y L + X R y − X R Y L } {\displaystyle {\begin{aligned}xy&=\{X_{L}\mid X_{R}\}\{Y_{L}\mid Y_{R}\}\\&=\left\{X_{L}y+xY_{L}-X_{L}Y_{L},X_{R}y+xY_{R}-X_{R}Y_{R}\mid X_{L}y+xY_{R}-X_{L}Y_{R},xY_{L}+X_{R}y-X_{R}Y_{L}\right\}\\\end{aligned}}}
The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression X R y + x Y R − X R Y R {\textstyle X_{R}y+xY_{R}-X_{R}Y_{R}} that appears in the left set of the product of x and y. This is understood as { x ′ y + x y ′ − x ′ y ′ : x ′ ∈ X R , y ′ ∈ Y R } {\textstyle \left\{x'y+xy'-x'y':x'\in X_{R},~y'\in Y_{R}\right\}} , the set of numbers generated by picking all possible combinations of members of X R {\textstyle X_{R}} and Y R {\textstyle Y_{R}} , and substituting them into the expression. For example, to show that the square of 1/2 is 1/4:
Division The definition of division is done in terms of the reciprocal and multiplication:
x y = x ⋅ 1 y {\displaystyle {\frac {x}{y}}=x\cdot {\frac {1}{y}}}
where
1 y = { 0 , 1 + ( y R − y ) ( 1 y ) L y R , 1 + ( y L − y ) ( 1 y ) R y L | 1 + ( y L − y ) ( 1 y ) L y L , 1 + ( y R − y ) ( 1 y ) R y R } {\displaystyle {\frac {1}{y}}=\left\{\left.0,{\frac {1+(y_{R}-y)\left({\frac {1}{y}}\right)_{L}}{y_{R}}},{\frac {1+\left(y_{L}-y\right)\left({\frac {1}{y}}\right)_{R}}{y_{L}}}\,\,\right|\,\,{\frac {1+(y_{L}-y)\left({\frac {1}{y}}\right)_{L}}{y_{L}}},{\frac {1+(y_{R}-y)\left({\frac {1}{y}}\right)_{R}}{y_{R}}}\right\}}
for positive y. Only positive yL are permitted in the formula, with any nonpositive terms being ignored (and yR are always positive). This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets of y, but also recursion in that the members of the left and right sets of 1/y itself. 0 is always a member of the left set of 1/y, and that can be used to find more terms in a recursive fashion. For example, if y = 3 = { 2 |}, then we know a left term of 1/3 will be 0. This in turn means 1 + (2 − 3)0/2 = 1/2 is a right term. This means
1 + ( 2 − 3 ) ( 1 2 ) 2 = 1 4 {\displaystyle {\frac {1+(2-3)\left({\frac {1}{2}}\right)}{2}}={\frac {1}{4}}}
is a left term. This means
1 + ( 2 − 3 ) ( 1 4 ) 2 = 3 8 {\displaystyle {\frac {1+(2-3)\left({\frac {1}{4}}\right)}{2}}={\frac {3}{8}}}
will be a right term. Continuing, this gives
1 3 = { 0 , 1 4 , 5 16 , … | 1 2 , 3 8 , … } {\displaystyle {\frac {1}{3}}=\left\{\left.0,{\frac {1}{4}},{\frac {5}{16}},\ldots \,\right|\,{\frac {1}{2}},{\frac {3}{8}},\ldots \right\}}
For negative y, 1/y is given by
1 y = − ( 1 − y ) {\displaystyle {\frac {1}{y}}=-\left({\frac {1}{-y}}\right)}
If y = 0, then 1/y is undefined.
Consistency It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:
Addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday n will eventually be expressed entirely in terms of operations on numbers with birthdays less than n; Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday n will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than n; As long as the operands are well-defined surreal number forms (each element of the left set is less than each element of the right set), the results are again well-defined surreal number forms; The operations can be extended to numbers (equivalence classes of forms): the result of negating x or adding or multiplying x and y will represent the same number regardless of the choice of form of x and y; and These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a field, with additive identity 0 = { | } and multiplicative identity 1 = { 0 | }. With these rules one can now verify that the numbers found in the first few generations were properly labeled. The construction rule is repeated to obtain more generations of surreals:
Arithmetic closure For each natural number (finite ordinal) n, all numbers generated in Sn are dyadic fractions, i.e., can be written as an irreducible fraction a/2b, where a and b are integers and 0 ≤ b < n. The set of all surreal numbers that are generated in some Sn for finite n may be denoted as S ∗ = ⋃ n ∈ N S n {\textstyle S_{*}=\bigcup _{n\in N}S_{n}} . One may form the three classes
S 0 = { 0 } S + = { x ∈ S ∗ : x > 0 } S − = { x ∈ S ∗ : x < 0 } {\displaystyle {\begin{aligned}S_{0}&=\{0\}\\S_{+}&=\{x\in S_{*}:x>0\}\\S_{-}&=\{x\in S_{*}:x<0\}\end{aligned}}}
of which S∗ is the union. No individual Sn is closed under addition and multiplication (except S0), but S∗ is; it is the subring of the rationals consisting of all dyadic fractions. There are infinite ordinal numbers β for which the set of surreal numbers with birthday less than β is closed under the different arithmetic operations. For any ordinal α, the set of surreal numbers with birthday less than β = ωα (using powers of ω) is closed under addition and forms a group; for birthday less than ωωα it is closed under multiplication and forms a ring; and for birthday less than an (ordinal) epsilon number εα it is closed under multiplicative inverse and forms a field. The latter sets are also closed under the exponential function as defined by Kruskal and Gonshor. However, it is always possible to construct a surreal number that is greater than any member of a set of surreals (by including the set on the left side of the constructor) and thus the collection of surreal numbers is a proper class. With their ordering and algebraic operations they constitute an ordered field, with the caveat that they do not form a set. In fact, it is a very special ordered field: the biggest one, in that every ordered field is a subfield of the surreal numbers. The class of all surreal numbers is denoted by the symbol N o {\textstyle \mathbb {No} } .
Infinity Define Sω as the set of all surreal numbers generated by the construction rule from subsets of