Surreal number

File:Surreal number tree.svg

A visualization of the surreal number tree.

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 Hausdorff introduced certain ordered sets called η_α-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 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.

Description

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.

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 the 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

{ 0 | } = 1

{ 1 | } = 2

{ 2 | } = 3

and

{ | 0 } = −1

{ | −1 } = −2

{ | −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 such as

{ 0 | 1 } = 1/2

{ 0 | 1/2 } = 1/4

{ 1/2 | 1 } = 3/4

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 { L_a | R_a }, where L_a is the set of all dyadic rationals less than a and R_a 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

{ 0, 1, 2, 3, ... | } = ω

{ 0 | 1, 1/2, 1/4, 1/8, ... } = 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.

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

A form { L | R } is numeric if the intersection of L and R is the empty set and each element of R is greater than every element of L, according to the order relation ≤ given by the comparison rule below.

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

Two numeric forms x and y are forms of the same number (lie in the same equivalence class) if and only if both x ≤ y and y ≤ x.

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 = { X_L | X_R } and y = { Y_L | Y_R }, x ≤ y if and only if both:

• There is no x_L ∈ X_L such that y ≤ x_L. That is, every element in the left part of x is strictly smaller than y.
• There is no y_R ∈ Y_R such that y_R ≤ 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 S_n is the set of all surreal numbers that are generated by the construction rule from subsets of ${\textstyle \bigcup _{i<n}S_{i}}$.

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 S_i with i < 0, the expression ${\textstyle \bigcup _{i<0}S_{i}}$ 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, S_n 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 S_i is also a valid form in S_n, all of the numbers in S_i also appear in S_n (as supersets of their representation in S_i). (The set union expression appears in our construction rule, rather than the simpler form S_n−1, so that the definition also makes sense when n is a limit ordinal.) Numbers in S_n that are a superset of some number in S_i are said to have been inherited from generation i. The smallest value of α for which a given surreal number appears in S_α 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:

{ | −1 } = { | −1, 0 } = { | −1, 1 } = { | −1, 0, 1 }

< { | 0 } = { | 0, 1 }

< { −1 | 0 } = { −1 | 0, 1 }

< { | } = { −1 | } = { | 1 } = { −1 | 1 }

< { 0 | 1 } = { −1, 0 | 1 }

< { 0 | } = { −1, 0 | }

< { 1 | } = { 0, 1 | } = { −1, 1 | } = { −1, 0, 1 | }

Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow:

1. 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.
2. 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).
3. The equivalence class of a number depends only on 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 number halfway between 0 and 1" and "the number halfway 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:

−2 < −1 < −1/2 < 0 < 1/2 < 1 < 2.

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 S_i 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 exactly one number c with this least i as its birthday that lies between L and R; x is a form of this c. In other words, it lies in the equivalence class in S_n 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 = { X_L | X_R } and y = { Y_L | Y_R } are defined by three recursive formulas.

Negation

Negation of a given number x = { X_L | X_R } is defined by

$-x=-\{X_{L}\mid X_{R}\}=\{-X_{R}\mid -X_{L}\},$

where the negation of a set S of numbers is given by the set of the negated elements of S:

$-S=\{-s:s\in S\}.$

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 only makes sense 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 X_L and X_R are drawn from generations earlier than that in which the form x first occurs, and observing the special case:

$-0=-\{{}\mid {}\}=\{{}\mid {}\}=0.$

Addition

The definition of addition is also a recursive formula:

$x+y=\{X_{L}\mid X_{R}\}+\{Y_{L}\mid Y_{R}\}=\{X_{L}+y,x+Y_{L}\mid X_{R}+y,x+Y_{R}\},$

where

$X+y=\{x+y:x\in X\},x+Y=\{x+y:y\in Y\}$.

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:

$0+0=\{{}\mid {}\}+\{{}\mid {}\}=\{{}\mid {}\}=0$

$x+0=x+\{{}\mid {}\}=\{X_{L}+0\mid X_{R}+0\}=\{X_{L}\mid X_{R}\}=x$

$0+y=\{\mid <!--\; \mid \; -->\}+y=\{0+Y_L$