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Surreal number

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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.Template:Efn 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.<ref name=bajnok>Template:Cite book</ref> 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 conceptEdit

Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers.<ref name="O'Connor">Template:Citation</ref> 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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 [[η set|Template:Math-sets]] for ordinals Template:Mvar 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 Template:Mvar and, in 1987, he showed that taking Template:Mvar to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.<ref>Template:Citation</ref>

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.<ref name="Alling1985">Template:Citation</ref>

DescriptionEdit

NotationEdit

In the context of surreal numbers, an ordered pair of sets Template:Mvar and Template:Mvar, which is written as Template:Math in many other mathematical contexts, is instead written Template:Math including the extra space adjacent to each brace. When a set is empty, it is often simply omitted. When a set is explicitly described by its elements, the pair of braces that encloses the list of elements is often omitted. When a union of sets is taken, the operator that represents that is often a comma. For example, instead of Template:Math, which is common notation in other contexts, we typically write Template:Math.

Outline of constructionEdit

In the Conway construction,<ref name="Con01">Template:Cite book</ref> the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers Template:Mvar and Template:Mvar, Template:Math or Template:Math. (Both may hold, in which case Template:Mvar and Template:Mvar are equivalent and denote the same number.) Each number is formed from an ordered pair of subsets of numbers already constructed: given subsets Template:Mvar and Template:Mvar of numbers such that all the members of Template:Mvar are strictly less than all the members of Template:Mvar, then the pair Template:Math represents a number intermediate in value between all the members of Template:Mvar and all the members of Template:Mvar.

Different subsets may end up defining the same number: Template:Math and Template:Math may define the same number even if Template:Math and Template:Math. (A similar phenomenon occurs when rational numbers are defined as quotients of integers: Template:Sfrac and Template:Sfrac are different representations of the same rational number.) So strictly speaking, the surreal numbers are equivalence classes of representations of the form Template:Math 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: Template:Math. This representation, where Template:Mvar and Template:Mvar are both empty, is called 0. Subsequent stages yield forms like

Template:Math
Template:Math
Template:Math

and

Template:Math
Template:Math
Template:Math

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

Template:Math
Template:Math
Template:Math

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 Template:Mvar can be represented by Template:Math where Template:Math is the set of all dyadic rationals less than Template:Mvar and Template:Math is the set of all dyadic rationals greater than Template:Mvar (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.

There are also representations like

Template:Math
Template:Math

where Template:Mvar is a transfinite number greater than all integers and Template:Mvar 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 Template:Math or Template:Math and so forth.

ConstructionEdit

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.

FormsEdit

A form is a pair of sets of surreal numbers, called its left set and its right set. A form with left set Template:Mvar and right set Template:Mvar is written Template:Math. When Template:Mvar and Template:Mvar 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 Template:Math with both left and right set empty is also written Template:Math.

Numeric forms and their equivalence classesEdit

Construction rule

A form Template:Math is numeric if the intersection of Template:Mvar and Template:Mvar is the empty set and each element of Template:Mvar is greater than every element of Template:Mvar, 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 Template:Mvar and Template:Mvar are forms of the same number (lie in the same equivalence class) if and only if both Template:Math and Template:Math.

An ordering relationship must be antisymmetric, i.e., it must have the property that Template:Math (i. e., Template:Math and Template:Math are both true) only when Template:Mvar and Template:Mvar 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 Template:Math is labeled 0; in other words, Template:Math is a form of the surreal number 0.

OrderEdit

The recursive definition of surreal numbers is completed by defining comparison:

Given numeric forms Template:Math and Template:Math, Template:Math if and only if both:

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.

InductionEdit

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 ruleEdit

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 Template:Math with Template:Math, 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 Template:Math consists of a single surreal form Template:Math lying in a single equivalence class 0.

For every finite ordinal number Template:Mvar, Template:Math 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 Template:Math (the form Template:Math is non-numeric because Template:Math). The equivalence class containing Template:Nowrap is labeled 1 and the equivalence class containing Template:Nowrap 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 Template:Math, since every valid form in Template:Math is also a valid form in Template:Math, all of the numbers in Template:Math also appear in Template:Math (as supersets of their representation in Template:Math). (The set union expression appears in our construction rule, rather than the simpler form Template:Math, so that the definition also makes sense when Template:Mvar is a limit ordinal.) Numbers in Template:Math that are a superset of some number in Template:Math are said to have been inherited from generation Template:Mvar. The smallest value of Template:Mvar for which a given surreal number appears in Template:Math 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:

Template:Math
Template:Math
Template:Math
Template:Math
Template:Math
Template:Math
Template:Math

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

  1. Template:Math contains four new surreal numbers. Two contain extremal forms: Template:Math contains all numbers from previous generations in its right set, and Template:Math 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 Template:Mvar 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 Template:Mvar from previous generations into a left set (all numbers less than Template:Mvar) and a right set (all numbers greater than Template:Mvar).
  3. 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 Template:Math and Template:Math 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 Template:Math and Template:Math are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled Template:Sfrac and −Template:Sfrac. These labels will also be justified by the rules for surreal addition and multiplication below.

The equivalence classes at each stage Template:Mvar of induction may be characterized by their Template:Mvar-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:

Template:Math.

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 Template:Math is therefore equivalent to Template:Math; one can establish that these are forms of 3 by using the birthday property, which is a consequence of the rules above.

Birthday propertyEdit

A form Template:Math occurring in generation Template:Mvar represents a number inherited from an earlier generation Template:Math if and only if there is some number in Template:Math that is greater than all elements of Template:Mvar and less than all elements of the Template:Mvar. (In other words, if Template:Mvar and Template:Mvar are already separated by a number created at an earlier stage, then Template:Mvar does not represent a new number but one already constructed.) If Template:Mvar represents a number from any generation earlier than Template:Mvar, there is a least such generation Template:Mvar, and exactly one number Template:Mvar with this least Template:Mvar as its birthday that lies between Template:Mvar and Template:Mvar; Template:Mvar is a form of this Template:Mvar. In other words, it lies in the equivalence class in Template:Math that is a superset of the representation of Template:Mvar in generation Template:Mvar.

ArithmeticEdit

The addition, negation (additive inverse), and multiplication of surreal number forms Template:Math and Template:Math are defined by three recursive formulas.

NegationEdit

Negation of a given number Template:Math is defined by <math display=block>-x = - \{ X_L \mid X_R \} = \{ -X_R \mid -X_L \},</math> where the negation of a set Template:Mvar of numbers is given by the set of the negated elements of Template:Mvar: <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 Template:Mvar, 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 Template:Math and Template:Math are drawn from generations earlier than that in which the form Template:Mvar first occurs, and observing the special case: <math display=block>-0 = - \{ {}\mid{} \} = \{ {}\mid{} \} = 0.</math>

AdditionEdit

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:

Template:Math,

which by the birthday property is a form of 1. This justifies the label used in the previous section.

SubtractionEdit

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>

MultiplicationEdit

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 Template:Mvar and Template:Mvar. 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 Template:Sfrac is Template:Sfrac:

Template:Math.

DivisionEdit

The definition of division is done in terms of the reciprocal and multiplication:

<math display=block>\frac xy = x \cdot \frac 1y</math>

where<ref name="Con01" />Template:Rp

<math display=block>\frac 1y = \left\{\left.0, \frac{1+(y_R-y)\left(\frac1y\right)_L}{y_R}, \frac{1+\left(y_L-y\right)\left(\frac1y\right)_R}{y_L} \,\,\right|\,\, \frac{1+(y_L-y)\left(\frac1y\right)_L}{y_L}, \frac{1+(y_R-y)\left(\frac1y\right)_R}{y_R} \right\}</math>

for positive Template:Mvar. Only positive Template:Math are permitted in the formula, with any nonpositive terms being ignored (and Template:Math 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 Template:Mvar, but also recursion in that the members of the left and right sets of Template:Math itself. 0 is always a member of the left set of Template:Math, and that can be used to find more terms in a recursive fashion. For example, if Template:Math}, then we know a left term of Template:Sfrac will be 0. This in turn means Template:Math is a right term. This means <math display=block>\frac{1+(2-3)\left(\frac12\right)}2=\frac14</math> is a left term. This means <math display=block>\frac{1+(2-3)\left(\frac14\right)}2 = \frac 38</math> will be a right term. Continuing, this gives <math display=block>\frac13 = \left\{\left. 0, \frac14, \frac5{16}, \ldots \,\right|\, \frac12, \frac38, \ldots\right\}</math>

For negative Template:Mvar, Template:Math is given by <math display=block>\frac1y=-\left(\frac1{-y}\right)</math>

If Template:Math, then Template:Math is undefined.

ConsistencyEdit

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 Template:Mvar will eventually be expressed entirely in terms of operations on numbers with birthdays less than Template:Mvar;
  • Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday Template:Mvar will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than Template:Mvar;
  • 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 Template:Mvar or adding or multiplying Template:Mvar and Template:Mvar will represent the same number regardless of the choice of form of Template:Mvar and Template:Mvar; and
  • These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a field, with additive identity Template:Math and multiplicative identity Template:Math.

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:

Template:Math
Template:Math
Template:Math
Template:Math
Template:Math

Arithmetic closureEdit

For each natural number (finite ordinal) Template:Mvar, all numbers generated in Template:Math are dyadic fractions, i.e., can be written as an irreducible fraction Template:Math, where Template:Mvar and Template:Mvar are integers and Template:Math.

The set of all surreal numbers that are generated in some Template:Math for finite Template:Mvar may be denoted as <math display=inline>S_* = \bigcup_{n \in N} S_n</math>. One may form the three classes <math display=block>\begin{align} S_{0} &= \{ 0 \} \\ S_{+} &= \{ x \in S_*: x > 0 \} \\ S_{-} &= \{ x \in S_*: x < 0 \} \end{align}</math> of which Template:Math is the union. No individual Template:Math is closed under addition and multiplication (except Template:Math), but Template:Math is; it is the subring of the rationals consisting of all dyadic fractions.

There are infinite ordinal numbers Template:Mvar for which the set of surreal numbers with birthday less than Template:Mvar is closed under the different arithmetic operations.<ref name=vdDE2001>Template:Cite journal</ref> For any ordinal Template:Mvar, the set of surreal numbers with birthday less than Template:Math (using [[#Powers of ω|powers of Template:Mvar]]) is closed under addition and forms a group; for birthday less than Template:Mvar it is closed under multiplication and forms a ring;Template:Efn and for birthday less than an (ordinal) epsilon number Template:Mvar 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.<ref name=vdDE2001 /><ref name=G1986>Template:Cite book</ref>Template:Rp<ref name=vdDE2001 />

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 the biggest ordered field, in that every ordered field is a subfield of the surreal numbers.<ref name=bajnok/> The class of all surreal numbers is denoted by the symbol <math display=inline>\mathbb{No}</math>.

InfinityEdit

Define Template:Math as the set of all surreal numbers generated by the construction rule from subsets of Template:Math. (This is the same inductive step as before, since the ordinal number Template:Mvar is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can be performed only in a set theory that allows such a union.) A unique infinitely large positive number occurs in Template:Math: <math display=block>\omega = \{ S_* \mid{} \} = \{ 1, 2, 3, 4, \ldots \mid{} \}.</math> Template:Math also contains objects that can be identified as the rational numbers. For example, the Template:Mvar-complete form of the fraction Template:Sfrac is given by: <math display=block>\tfrac{1} {3} = \{ y \in S_*: 3 y < 1 \mid y \in S_*: 3 y > 1 \}.</math> The product of this form of Template:Sfrac with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.

Not only do all the rest of the rational numbers appear in Template:Math; the remaining finite real numbers do too. For example, <math display=block>\pi = \left\{ 3, \tfrac{25}{8},\tfrac{201}{64}, \ldots \mid 4, \tfrac{7}{2}, \tfrac{13}{4}, \tfrac{51}{16},\ldots \right\}.</math>

The only infinities in Template:Math are Template:Mvar and Template:Math; but there are other non-real numbers in Template:Math among the reals. Consider the smallest positive number in Template:Math: <math display=block>\varepsilon = \{ S_- \cup S_0 \mid S_+ \} = \left\{ 0 \mid 1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \ldots \right\} = \{ 0 \mid y \in S_* : y > 0 \}</math> This number is larger than zero but less than all positive dyadic fractions. It is therefore an infinitesimal number, often labeled Template:Mvar. The Template:Mvar-complete form of Template:Mvar (respectively Template:Math) is the same as the Template:Mvar-complete form of 0, except that 0 is included in the left (respectively right) set. The only "pure" infinitesimals in Template:Math are Template:Mvar and its additive inverse Template:Math; adding them to any dyadic fraction Template:Mvar produces the numbers Template:Math, which also lie in Template:Math.

One can determine the relationship between Template:Mvar and Template:Mvar by multiplying particular forms of them to obtain:

Template:Math.

This expression is well-defined only in a set theory which permits transfinite induction up to Template:Math. In such a system, one can demonstrate that all the elements of the left set of Template:Math are positive infinitesimals and all the elements of the right set are positive infinities, and therefore Template:Math is the oldest positive finite number, 1. Consequently, Template:Math. Some authors systematically use Template:Math in place of the symbol Template:Mvar.

Contents of SωEdit

Given any Template:Math in Template:Math, exactly one of the following is true:

Template:Math is not an algebraic field, because it is not closed under arithmetic operations; consider Template:Math, whose form <math display=block>\omega + 1 = \{ 1, 2, 3, 4, ... \mid {} \} + \{ 0 \mid{} \} = \{ 1, 2, 3, 4, \ldots, \omega \mid {} \}</math> does not lie in any number in Template:Math. The maximal subset of Template:Math that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities Template:Math, the infinitesimals Template:Math, and the infinitesimal neighbors Template:Math of each nonzero dyadic fraction Template:Mvar.

This construction of the real numbers differs from the Dedekind cuts of standard analysis in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in Template:Math with its forms in previous generations. (The Template:Mvar-complete forms of real elements of Template:Math are in one-to-one correspondence with the reals obtained by Dedekind cuts, under the proviso that Dedekind reals corresponding to rational numbers are represented by the form in which the cut point is omitted from both left and right sets.) The rationals are not an identifiable stage in the surreal construction; they are merely the subset Template:Mvar of Template:Math containing all elements Template:Mvar such that Template:Math for some Template:Mvar and some nonzero Template:Mvar, both drawn from Template:Math. By demonstrating that Template:Mvar is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element of Template:Mvar is reachable from Template:Math by a finite series (no longer than two, actually) of arithmetic operations including multiplicative inversion, one can show that Template:Mvar is strictly smaller than the subset of Template:Math identified with the reals.

The set Template:Math has the same cardinality as the real numbers Template:Mvar. This can be demonstrated by exhibiting surjective mappings from Template:Math to the closed unit interval Template:Mvar of Template:Mvar and vice versa. Mapping Template:Math onto Template:Mvar is routine; map numbers less than or equal to Template:Mvar (including Template:Math) to 0, numbers greater than or equal to Template:Math (including Template:Mvar) to 1, and numbers between Template:Mvar and Template:Math to their equivalent in Template:Mvar (mapping the infinitesimal neighbors Template:Math of each dyadic fraction Template:Mvar, along with Template:Mvar itself, to Template:Mvar). To map Template:Mvar onto Template:Math, map the (open) central third (Template:Sfrac, Template:Sfrac) of Template:Mvar onto Template:Math; the central third (Template:Sfrac, Template:Sfrac) of the upper third to Template:Math; and so forth. This maps a nonempty open interval of Template:Mvar onto each element of Template:Math, monotonically. The residue of Template:Mvar consists of the Cantor set Template:Math, each point of which is uniquely identified by a partition of the central-third intervals into left and right sets, corresponding precisely to a form Template:Math in Template:Math. This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday Template:Mvar.

Transfinite inductionEdit

Continuing to perform transfinite induction beyond Template:Math produces more ordinal numbers Template:Mvar, each represented as the largest surreal number having birthday Template:Mvar. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is Template:Math. There is another positive infinite number in generation Template:Math:

Template:Math.

The surreal number Template:Math is not an ordinal; the ordinal Template:Math is not the successor of any ordinal. This is a surreal number with birthday Template:Math, which is labeled Template:Math on the basis that it coincides with the sum of Template:Math and Template:Math. Similarly, there are two new infinitesimal numbers in generation Template:Math:

Template:Math and
Template:Math.

At a later stage of transfinite induction, there is a number larger than Template:Math for all natural numbers Template:Math:

Template:Math

This number may be labeled Template:Math both because its birthday is Template:Math (the first ordinal number not reachable from Template:Math by the successor operation) and because it coincides with the surreal sum of Template:Math and Template:Math; it may also be labeled Template:Math because it coincides with the product of Template:Math and Template:Math. It is the second limit ordinal; reaching it from Template:Math via the construction step requires a transfinite induction on <math display=block>\bigcup_{k < \omega} S_{\omega + k}</math> This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.

Note that the conventional addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals Template:Math equals Template:Math, but the surreal sum is commutative and produces Template:Math. The addition and multiplication of the surreal numbers associated with ordinals coincides with the natural sum and natural product of ordinals.

Just as Template:Math is bigger than Template:Math for any natural number Template:Math, there is a surreal number Template:Math that is infinite but smaller than Template:Math for any natural number Template:Math. That is, Template:Math is defined by

Template:Math

where on the right hand side the notation Template:Math is used to mean Template:Math. It can be identified as the product of Template:Math and the form Template:Math of Template:Math. The birthday of Template:Math is the limit ordinal Template:Math.

Powers of ω and the Conway normal formEdit

To classify the "orders" of infinite and infinitesimal surreal numbers, also known as archimedean classes, Conway associated to each surreal number Template:Mvar the surreal number

where Template:Mvar and Template:Mvar range over the positive real numbers. If Template:Math then Template:Math is "infinitely greater" than Template:Math, in that it is greater than Template:Math for all real numbers Template:Mvar. Powers of Template:Mvar also satisfy the conditions

so they behave the way one would expect powers to behave.

Each power of Template:Mvar also has the redeeming feature of being the simplest surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal number Template:Mvar there will always exist some positive real number Template:Mvar and some surreal number Template:Mvar so that Template:Math is "infinitely smaller" than Template:Mvar. The exponent Template:Mvar is the "base Template:Mvar logarithm" of Template:Mvar, defined on the positive surreals; it can be demonstrated that Template:Math maps the positive surreals onto the surreals and that

Template:Math.

This gets extended by transfinite induction so that every surreal number has a "normal form" analogous to the Cantor normal form for ordinal numbers. This is the Conway normal form: Every surreal number Template:Mvar may be uniquely written as

Template:Math,

where every Template:Math is a nonzero real number and the Template:Maths form a strictly decreasing sequence of surreal numbers. This "sum", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number. (Zero corresponds of course to the case of an empty sequence, and is the only surreal number with no leading exponent.)

Looked at in this manner, the surreal numbers resemble a power series field, except that the decreasing sequences of exponents must be bounded in length by an ordinal and are not allowed to be as long as the class of ordinals. This is the basis for the formulation of the surreal numbers as a Hahn series.

Gaps and continuityEdit

In contrast to the real numbers, a (proper) subset of the surreal numbers does not have a least upper (or lower) bound unless it has a maximal (minimal) element. Conway defines<ref name=Con01/> a gap as Template:Math such that every element of Template:Math is less than every element of Template:Math, and <math display=inline> L \cup R = \mathbb{No}</math>; this is not a number because at least one of the sides is a proper class. Though similar, gaps are not quite the same as Dedekind cuts,Template:Efn but we can still talk about a completion <math display=inline>\mathbb{No}_\mathfrak{D}</math> of the surreal numbers with the natural ordering which is a (proper class-sized) linear continuum.<ref name=RSS15>Template:Cite arXiv</ref>

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, Template:Mvar "is" the set of ordinals smaller than Template:Mvar, and we can use this equivalence to write Template:Nowrap 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,Template:Efn <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.<ref name=RSS15/> An equivalent of Cauchy sequences can be defined as well, although they have to be indexed by the class of ordinals; these will always converge, but the limit may be either a number or a gap that can be expressed as <math display=block>\sum_{\alpha\in\mathbb{No}} r_\alpha \omega^{a_\alpha}</math> with Template:Math decreasing and having no lower bound in <math display=inline>\mathbb{No}</math>. (All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as Template:Math and <math display=inline>\mathbb{On}</math>).<ref name=RSS15/>

Exponential functionEdit

Based on unpublished work by Kruskal, a construction (by transfinite induction) that extends the real exponential function Template:Math (with base Template:Mvar) to the surreals was carried through by Gonshor.<ref name=G1986 />Template:Rp

Other exponentialsEdit

The [[#Powers of ω|powers of Template:Mvar]] function is also an exponential function, but does not have the properties desired for an extension of the function on the reals. It will, however, be needed in the development of the base-Template:Mvar exponential, and it is this function that is meant whenever the notation Template:Mvar is used in the following.

When Template:Mvar is a dyadic fraction, the power function <math display=inline>x \in \mathbb{No}</math>, Template:Math may be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively. Its values are completely determined by the basic relation Template:Math, and where defined it necessarily agrees with any other exponentiation that can exist.

Basic inductionEdit

The induction steps for the surreal exponential are based on the series expansion for the real exponential, <math display=block>\exp x = \sum_{n\ge 0} \frac{x^n}{n!}</math> more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For Template:Mvar positive these are denoted Template:Math and include all partial sums; for Template:Mvar negative but finite, Template:Math denotes the odd steps in the series starting from the first one with a positive real part (which always exists). For Template:Mvar negative infinite the odd-numbered partial sums are strictly decreasing and the Template:Math notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.

The relations that hold for real Template:Math are thenTemplate:UblandTemplate:Ubland this can be extended to the surreals with the definition

<math display=block>\exp z = \{0, \exp z_L \cdot [z-z_L]_n, \exp z_R\cdot[z-z_R]_{2n+1} \mid \exp z_R/[z_R-z]_n, \exp z_L/[z_L-z]_{2n+1} \}.</math>

This is well-defined for all surreal arguments (the value exists and does not depend on the choice of Template:Mvar and Template:Mvar).

ResultsEdit

Using this definition, the following hold:Template:Efn

ExamplesEdit

The surreal exponential is essentially given by its behaviour on positive powers of Template:Mvar, i.e., the function Template:Tmath, combined with well-known behaviour on finite numbers. Only examples of the former will be given. In addition, Template:Tmath holds for a large part of its range, for instance for any finite number with positive real part and any infinite number that is less than some iterated power of Template:Mvar (Template:Mvar for some number of levels).

ExponentiationEdit

A general exponentiation can be defined as Template:Nowrap, giving an interpretation to expressions like Template:Nowrap. Again it is essential to distinguish this definition from the "powers of Template:Mvar" function, especially if Template:Mvar may occur as the base.

Surcomplex numbersEdit

A surcomplex number is a number of the form Template:Math, where Template:Mvar and Template:Mvar are surreal numbers and Template:Mvar is the square root of Template:Math.<ref>Surreal vectors and the game of Cutblock, James Propp, August 22, 1994.</ref><ref name="Alling">Template:Cite book</ref> 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.<ref name=Con01/>Template:Rp

GamesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The definition of surreal numbers contained one restriction: each element of Template:Mvar must be strictly less than each element of Template:Mvar. If this restriction is dropped we can generate a more general class known as games. All games are constructed according to this rule:

Construction rule
If Template:Mvar and Template:Mvar are two sets of games then Template:Math is a game.

Addition, negation, and comparison are all defined the same way for both surreal numbers and games.

Every surreal number is a game, but not all games are surreal numbers, e.g. the game [[star (game theory)|Template:Math]] is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a field, but the class of games does not. The surreals have a total order: given any two surreals, they are either equal, or one is greater than the other. The games have only a partial order: there exist pairs of games that are neither equal, greater than, nor less than each other. Each surreal number is either positive, negative, or zero. Each game is either positive, negative, zero, or fuzzy (incomparable with zero, such as Template:Math).

A move in a game involves the player whose move it is choosing a game from those available in Template:Mvar (for the left player) or Template:Mvar (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a fuzzy game for the first player to move.

If Template:Mvar, Template:Mvar, and Template:Mvar are surreals, and Template:Math, then Template:Math. However, if Template:Mvar, Template:Mvar, and Template:Mvar are games, and Template:Math, then it is not always true that Template:Math. Note that "Template:Math" here means equality, not identity.

Application to combinatorial game theoryEdit

The surreal numbers were originally motivated by studies of the game Go,<ref name="O'Connor"/> and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized Game for the mathematical object Template:Math, and the lowercase game for recreational games like Chess or Go.

We consider games with these properties:

  • Two players (named Left and Right)
  • Deterministic (the game at each step will completely depend on the choices the players make, rather than a random factor)
  • No hidden information (such as cards or tiles that a player hides)
  • Players alternate taking turns (the game may or may not allow multiple moves in a turn)
  • Every game must end in a finite number of moves
  • As soon as there are no legal moves left for a player, the game ends, and that player loses

For most games, the initial board position gives no great advantage to either player. As the game progresses and one player starts to win, board positions will occur in which that player has a clear advantage. For analyzing games, it is useful to associate a Game with every board position. The value of a given position will be the Game Template:Math, where Template:Mvar is the set of values of all the positions that can be reached in a single move by Left. Similarly, Template:Mvar is the set of values of all the positions that can be reached in a single move by Right.

The zero Game (called Template:Math) is the Game where Template:Mvar and Template:Mvar are both empty, so the player to move next (Template:Mvar or Template:Mvar) immediately loses. The sum of two Games Template:Math and Template:Math is defined as the Game Template:Math where the player to move chooses which of the Games to play in at each stage, and the loser is still the player who ends up with no legal move. One can imagine two chess boards between two players, with players making moves alternately, but with complete freedom as to which board to play on. If Template:Mvar is the Game Template:Math, Template:Math is the Game Template:Math, i.e. with the role of the two players reversed. It is easy to show Template:Math for all Games Template:Mvar (where Template:Math is defined as Template:Math).

This simple way to associate Games with games yields a very interesting result. Suppose two perfect players play a game starting with a given position whose associated Game is Template:Mvar. We can classify all Games into four classes as follows:

More generally, we can define Template:Math as Template:Math, and similarly for Template:Math, Template:Math and Template:Math.

The notation Template:Math means that Template:Mvar and Template:Mvar are incomparable. Template:Math is equivalent to Template:Math, i.e. that Template:Math, Template:Math and Template:Math are all false. Incomparable games are sometimes said to be confused with each other, because one or the other may be preferred by a player depending on what is added to it. A game confused with zero is said to be fuzzy, as opposed to positive, negative, or zero. An example of a fuzzy game is star (*).

Sometimes when a game nears the end, it will decompose into several smaller games that do not interact, except in that each player's turn allows moving in only one of them. For example, in Go, the board will slowly fill up with pieces until there are just a few small islands of empty space where a player can move. Each island is like a separate game of Go, played on a very small board. It would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game. This doesn't appear to be easy to do. For example, there might be two subgames where whoever moves first wins, but when they are combined into one big game, it is no longer the first player who wins. Fortunately, there is a way to do this analysis. The following theorem can be applied:

If a big game decomposes into two smaller games, and the small games have associated Games of Template:Mvar and Template:Mvar, then the big game will have an associated Game of Template:Math.

A game composed of smaller games is called the disjunctive sum of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the disjunctive sum of the addends.

Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their disjunctive sum. From this he invented the concept of a Game and the addition operator for it. From there he moved on to developing a definition of negation and comparison. Then he noticed that a certain class of Games had interesting properties; this class became the surreal numbers. Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.

Alternative realizationsEdit

Alternative approaches to the surreal numbers complement the original exposition by Conway in terms of games.

Sign expansionEdit

DefinitionsEdit

In what is now called the sign-expansion or sign-sequence of a surreal number, a surreal number is a function whose domain is an ordinal and whose codomain is Template:Math.<ref name=G1986 />Template:Rp This notion has been introduced by Conway himself in the equivalent formulation of L-R sequences.<ref name="Con01" />

Define the binary predicate "simpler than" on numbers by: Template:Mvar is simpler than Template:Mvar if Template:Mvar is a proper subset of Template:Mvar, i.e. if Template:Math and Template:Math for all Template:Math.

For surreal numbers define the binary relation Template:Math to be lexicographic order (with the convention that "undefined values" are greater than Template:Math and less than Template:Math). So Template:Math if one of the following holds:

Equivalently, let Template:Math, so that Template:Math if and only if Template:Math. Then, for numbers Template:Mvar and Template:Mvar, Template:Math if and only if one of the following holds:

For numbers Template:Mvar and Template:Mvar, Template:Math if and only if Template:Math, and Template:Math if and only if Template:Math. Also Template:Math if and only if Template:Math.

The relation Template:Math is transitive, and for all numbers Template:Mvar and Template:Mvar, exactly one of Template:Math, Template:Math, Template:Math, holds (law of trichotomy). This means that Template:Math is a linear order (except that Template:Math is a proper class).

For sets of numbers Template:Mvar and Template:Mvar such that Template:Math, there exists a unique number Template:Mvar such that

Furthermore, Template:Mvar is constructible from Template:Mvar and Template:Mvar by transfinite induction. Template:Mvar is the simplest number between Template:Mvar and Template:Mvar. Let the unique number Template:Mvar be denoted by Template:Math.

For a number Template:Mvar, define its left set Template:Math and right set Template:Math by

then Template:Math.

One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's original realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals.

However, similar definitions can be made that eliminate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule Template:Math and whose range is Template:Math. "Simpler than" is very simply defined now: Template:Mvar is simpler than Template:Mvar if Template:Math. The total ordering is defined by considering Template:Mvar and Template:Mvar as sets of ordered pairs (as a function is normally defined): Either Template:Math, or else the surreal number Template:Math is in the domain of Template:Mvar or the domain of Template:Mvar (or both, but in this case the signs must disagree). We then have Template:Math if Template:Math or Template:Math (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of Template:MathTemplate:Hsp in order of simplicity (i.e., inclusion), and then write down the signs that Template:Math assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is Template:Math.

Addition and multiplicationEdit

The sum Template:Math of two numbers Template:Mvar and Template:Mvar is defined by induction on Template:Math and Template:Math by Template:Math, where

The additive identity is given by the number Template:Math, i.e. the number Template:Math is the unique function whose domain is the ordinal Template:Math, and the additive inverse of the number Template:Mvar is the number Template:Math, given by Template:Math, and, for Template:Math, Template:Math if Template:Math, and Template:Math if Template:Math.

It follows that a number Template:Mvar is positive if and only if Template:Math and Template:Math, and Template:Mvar is negative if and only if Template:Math and Template:Math.

The product Template:Mvar of two numbers, Template:Mvar and Template:Mvar, is defined by induction on Template:Math and Template:Math by Template:Math, where

The multiplicative identity is given by the number Template:Math, i.e. the number Template:Math has domain equal to the ordinal Template:Math, and Template:Math.

Correspondence with Conway's realizationEdit

The map from Conway's realization to sign expansions is given by Template:Math, where Template:Math and Template:Math.

The inverse map from the alternative realization to Conway's realization is given by Template:Math, where Template:Math and Template:Math.

Axiomatic approachEdit

In another approach to the surreals, given by Alling,<ref name="Alling" /> explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the axiomatic approach to the reals, these axioms guarantee uniqueness up to isomorphism.

A triple <math display=inline>\langle \mathbb{No}, \mathrm{<}, b \rangle</math> is a surreal number system if and only if the following hold:

Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.

Given these axioms, Alling<ref name="Alling"/> derives Conway's original definition of Template:Math and develops surreal arithmetic.

Simplicity hierarchyEdit

A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich.<ref name="Ehr12">Template:Cite journal</ref> 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.<ref name="Ehr12" />

Hahn seriesEdit

Alling<ref name="Alling" />Template:Rp 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.

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., Template:Math. 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.

See alsoEdit

Template:Portal

NotesEdit

Template:Notelist

ReferencesEdit

Template:Reflist

Further readingEdit

  • Donald Knuth's original exposition: Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness, 1974, Template:Isbn. More information can be found at the book's official homepage (archived).
  • An update of the classic 1976 book defining the surreal numbers, and exploring their connections to games: John Conway, On Numbers And Games, 2nd ed., 2001, Template:Isbn.
  • An update of the first part of the 1981 book that presented surreal numbers and the analysis of games to a broader audience: Berlekamp, Conway, and Guy, Winning Ways for Your Mathematical Plays, vol. 1, 2nd ed., 2001, Template:Isbn.
  • Martin Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman & Co., 1989, Template:Isbn, Chapter 4. A non-technical overview; reprint of the 1976 Scientific American article.
  • Polly Shulman, "Infinity Plus One, and Other Surreal Numbers", Discover, December 1995.
  • A detailed treatment of surreal numbers: Norman L. Alling, Foundations of Analysis over Surreal Number Fields, 1987, Template:Isbn.
  • A treatment of surreals based on the sign-expansion realization: Harry Gonshor, An Introduction to the Theory of Surreal Numbers, 1986, Template:Isbn.
  • A detailed philosophical development of the concept of surreal numbers as a most general concept of number: Alain Badiou, Number and Numbers, New York: Polity Press, 2008, Template:Isbn (paperback), Template:Isbn (hardcover).
  • Template:Cite book The surreal numbers are studied in the context of homotopy type theory in section 11.6.

External linksEdit

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