Surreal number
Surreal number
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 <math display=inline>\bigcup_{i < n} S_i</math>.
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 <math display=inline>\bigcup_{i < 0} S_i</math> 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.
The equivalence classes at each stage of induction may be characterized by their -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 is therefore equivalent to ; 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 occurring in generation represents a number inherited from an earlier generation if and only if there is some number in that is greater than all elements of and less than all elements of the . (In other words, if and are already separated by a number created at an earlier stage, then does not represent a new number but one already constructed.) If represents a number from any generation earlier than , there is a least such generation , and between and lies exactly one number that has this least as its birthday. is a form of this . In other words, it lies in the equivalence class in that is a superset of the representation of in generation .
Arithmetic The addition, negation (additive inverse), and multiplication of surreal number forms and are defined by three recursive formulas.
Negation Negation of a given number is defined by <math display=block>-x = - \{ X_L \mid X_R \} = \{ -X_R \mid -X_L \},</math> where the negation of a set of numbers is given by the set of the negated elements of : <math display=block>-S = \{ -s: s \in S \}.</math>
This formula involves the negation of the surreal numbers appearing in the left and right sets of , 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 and are drawn from generations earlier than that in which the form first occurs, and observing the special case: <math display=block>-0 = - \{ {}\mid{} \} = \{ {}\mid{} \} = 0.</math>
Addition The definition of addition is also a recursive formula: <math display=block>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 \},</math> where
<math display=block>X + y = \{ x' + y: x' \in X \} , \quad x + Y = \{ x + y': y' \in Y \}</math>
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: <math display=block>0 + 0 = \{ {}\mid{} \} + \{ {}\mid{} \} = \{ {}\mid{} \} = 0</math> <math display=block>x + 0 = x + \{ {}\mid{} \} = \{ X_L + 0 \mid X_R + 0 \} = \{ X_L \mid X_R \} = x</math> <math display=block>0 + y = \{ {}\mid{} \} + y = \{ 0 + Y_L \mid 0 + Y_R \} = \{ Y_L \mid Y_R \} = y</math>
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: <math display=block>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 \}\,.</math>
Multiplication Multiplication can be defined recursively as well, beginning from the special cases involving 0, the multiplicative identity 1, and its additive inverse −1: <math display=block>\begin{align} xy & = \{ X_L \mid X_R \} \{ Y_L \mid Y_R \} \\ & = \left\{ X_L y + x Y_L - X_L Y_L, X_R y + x Y_R - X_R Y_R \mid X_L y + x Y_R - X_L Y_R, x Y_L + X_R y - X_R Y_L \right\} \\ \end{align}</math> The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression <math display=inline>X_R y + x Y_R - X_R Y_R</math> that appears in the left set of the product of and . This is understood as <math display=inline>\left\{ x' y + x y' - x' y' : x' \in X_R,~ y' \in Y_R \right\}</math>, the set of numbers generated by picking all possible combinations of members of <math display=inline>X_R</math> and <math display=inline>Y_R</math>, and substituting them into the expression.
For example, to show that the square of is :
Division The definition of division is done in terms of the reciprocal and multiplication:
<math display=block>\frac xy = x \cdot \frac 1y</math>
where For any ordinal , the set of surreal numbers with birthday less than (using powers of) is closed under addition and forms a group; for birthday less than it is closed under multiplication and forms a ring; and for birthday less than an (ordinal) epsilon number 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.
For instance there is no least positive infinite surreal, but the gap
<math display=block>\{ x : \exists n \in \mathbb N : x < n\mid x : \forall n\in \mathbb N : x > n \}</math>
is greater than all real numbers and less than all positive infinite surreals, and is thus the least upper bound of the reals in <math display=inline>\mathbb{No}_\mathfrak{D}</math>. Similarly the gap <math display=inline>\mathbb{On} = \{ \mathbb{No} \mid{} \}</math> is larger than all surreal numbers. (This is an esoteric pun: In the general construction of ordinals, "is" the set of ordinals smaller than , and we can use this equivalence to write in the surreals; <math display=inline>\mathbb{On}</math> denotes the class of ordinal numbers, and because <math display=inline>\mathbb{On}</math> is cofinal in <math display=inline>\mathbb{No}</math> we have <math display=inline> \{ \mathbb{No} \mid {} \} = \{ \mathbb{On} \mid {} \} = \mathbb{On}</math> by extension.)
With a bit of set-theoretic care, <math display=inline>\mathbb{No}</math> can be equipped with a topology where the open sets are unions of open intervals (indexed by proper sets) and continuous functions can be defined. The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory. The difference from the usual definition of a tree is that the set of ancestors of a vertex is well-ordered, but may not have a maximal element (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation. Ehrlich additionally constructed an isomorphism between Conway's maximal surreal number field and the maximal hyperreals in von Neumann–Bernays–Gödel set theory.
Hahn series Alling also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of Hahn series with real coefficients on the value group of surreal numbers themselves (the series representation corresponding to the normal form of a surreal number, as defined above). This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.
Note that the support of the Hahn series must be a set, not a proper class; for instance, the Hahn series <math>\omega^{-\alpha}</math> summed over all ordinals has no surreal counterpart.
This isomorphism makes the surreal numbers into a valued field where the valuation is the additive inverse of the exponent of the leading term in the Conway normal form, e.g., . The valuation ring then consists of the finite surreal numbers (numbers with a real and/or an infinitesimal part). The reason for the sign inversion is that the exponents in the Conway normal form constitute a reverse well-ordered set, whereas Hahn series are formulated in terms of (non-reversed) well-ordered subsets of the value group.