{"history":[],"article":{"id":"39a72a46-4637-4dfd-9f81-ba97406521cd","title":"Surreal number","slug":"surreal-number","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **surreal number** system is a [totally ordered](https://bliptext.com/articles/total-order) [proper class](https://bliptext.com/articles/proper-class) containing not only the [real numbers](https://bliptext.com/articles/real-number) but also [infinite](https://bliptext.com/articles/infinity) and [infinite](https://bliptext.com/articles/infinity)simal numbers, respectively larger or smaller in [absolute value](https://bliptext.com/articles/absolute-value) than any positive real number.\nResearch on the [Go endgame](https://bliptext.com/articles/go-endgame) by [John Horton Conway](https://bliptext.com/articles/john-horton-conway) led to the original definition and construction of surreal numbers.\nConway's construction was introduced in [Donald Knuth](https://bliptext.com/articles/donald-knuth)'s 1974 book **Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness**.\nThe surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an [ordered field](https://bliptext.com/articles/ordered-field).\nIf formulated in [von Neumann–Bernays–Gödel set theory](https://bliptext.com/articles/von-neumann-bernays-g-del-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](https://bliptext.com/articles/rational-function), the [Levi-Civita field](https://bliptext.com/articles/levi-civita-field), the [superreal numbers](https://bliptext.com/articles/superreal-number) (including the [hyperreal numbers](https://bliptext.com/articles/hyperreal-number)) can be realized as subfields of the surreals.\nThe surreals also contain all [transfinite](https://bliptext.com/articles/transfinite-number) [ordinal numbers](https://bliptext.com/articles/ordinal-number); the arithmetic on them is given by the [natural operations](https://bliptext.com/articles/ordinal-arithmetic).\nIt has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is [isomorphic](https://bliptext.com/articles/isomorphic) to the maximal class surreal field.\n# [A visualization of the surreal number tree. | upload.wikimedia.org/wikipedia/commons/4/49/Surreal///number///tree.svg]\n\n# History of the concept\nResearch on the [Go endgame](https://bliptext.com/articles/go-strategy-and-tactics) by [John Horton Conway](https://bliptext.com/articles/john-horton-conway) led to the original definition and construction of the surreal numbers.\nConway's construction was introduced in [Donald Knuth](https://bliptext.com/articles/donald-knuth)'s 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.\nIn his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers.\nConway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book [On Numbers and Games](https://bliptext.com/articles/on-numbers-and-games).\nA separate route to defining the surreals began in 1907, when [Hans Hahn](https://bliptext.com/articles/hans-hahn-mathematician) introduced [Hahn series](https://bliptext.com/articles/hahn-series) as a generalization of [formal power series](https://bliptext.com/articles/formal-power-series), and [Felix Hausdorff](https://bliptext.com/articles/felix-hausdorff) introduced certain ordered sets called [[η set|\nηα\n-sets]] for ordinals α and asked if it was possible to find a compatible ordered group or field structure.\nIn 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.\nIf the surreals are considered as 'just' a proper-class-sized real closed field, Alling's 1962 paper handles the case of [strongly inaccessible](https://bliptext.com/articles/inaccessible-cardinal) cardinals which can naturally be considered as proper classes by cutting off the [cumulative hierarchy of the universe](https://bliptext.com/articles/von-neumann-universe) one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense.\nThere 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.\nThis 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.\n\n# Description\n\n## Notation\nIn the context of surreal numbers, an [ordered pair](https://bliptext.com/articles/ordered-pair) of sets L and R, which is written as\n(L, R)\nin many other mathematical contexts, is instead written\nincluding the extra space adjacent to each brace.\nWhen a set is empty, it is often simply omitted.\nWhen a set is explicitly described by its elements, the pair of braces that encloses the list of elements is often omitted.\nWhen a union of sets is taken, the operator that represents that is often a comma.\nFor example, instead of\n(L1 ∪ L2 ∪, ∅)\n, which is common notation in other contexts, we typically write\n.\n\n## Outline of construction\nIn the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers a and b,\na ≤ b\nor\nb ≤ a\n. (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\nrepresents a number intermediate in value between all the members of L and all the members of R.\nDifferent subsets may end up defining the same number:\nand\nmay define the same number even if\nL ≠ L′\nand\nR ≠ R′\n. (A similar phenomenon occurs when [rational numbers](https://bliptext.com/articles/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](https://bliptext.com/articles/equivalence-class) of representations of the form\nthat designate the same number.\nIn the first stage of construction, there are no previously existing numbers so the only representation must use the empty set:\n. This representation, where L and R are both empty, is called 0.\nSubsequent stages yield forms like\n= 1\n= 2\n= 3\nand\n= −1\n= −2\n= −3\nThe integers are thus contained within the surreal numbers.\n(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).\nSimilarly, representations such as\n= 1⁄2\n= 1⁄4\n= 3⁄4\narise, so that the [dyadic rationals](https://bliptext.com/articles/dyadic-rational) (rational numbers whose denominators are powers of 2) are contained within the surreal numbers.\nAfter an infinite number of stages, infinite subsets become available, so that any [real number](https://bliptext.com/articles/real-number) a can be represented by\n,\nwhere\nLa\nis the set of all dyadic rationals less than a and\nRa\nis the set of all dyadic rationals greater than a (reminiscent of a [Dedekind cut](https://bliptext.com/articles/dedekind-cut)).\nThus the real numbers are also embedded within the surreals.\nThere are also representations like\n= ω\n= ε\nwhere ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number.\nMoreover, 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\n2ω\nor\nω − 1\nand so forth.\n\n# Construction\nSurreal numbers are [constructed inductively](https://bliptext.com/articles/inductive-definition) as [equivalence classes](https://bliptext.com/articles/equivalence-class) of [pairs](https://bliptext.com/articles/ordered-pair) 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.\nThe construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.\n\n## Forms\nA form is a pair of sets of surreal numbers, called its left set and its right set.\nA form with left set L and right set R is written\n. When L and R are given as lists of elements, the braces around them are omitted.\nEither or both of the left and right set of a form may be the empty set.\nThe form\nwith both left and right set empty is also written\n.\n\n## Numeric forms and their equivalence classes\n**Construction rule**\nis 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](https://bliptext.com/articles/order-theory) ≤ given by the comparison rule below.\nThe numeric forms are placed in equivalence classes; each such equivalence class is a surreal number.\nThe 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).\n**Equivalence rule**\nx ≤ y\nand\ny ≤ x\n.\nAn [ordering relationship](https://bliptext.com/articles/order-theory) must be [antisymmetric](https://bliptext.com/articles/antisymmetric-relation), i.e., it must have the property that\nx = y\n(i. e.,\nx ≤ y\nand\ny ≤ x\nare 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).\nThe equivalence class containing\nis labeled 0; in other words,\nis a form of the surreal number 0.\n\n## Order\nThe recursive definition of surreal numbers is completed by defining comparison:\nGiven numeric forms\nx =\nand\ny =\n,\nx ≤ y\nif and only if both:\nxL ∈ XL\nsuch that\ny ≤ xL\n. That is, every element in the left part of x is strictly smaller than y.\nyR ∈ YR\nsuch that\nyR ≤ x\n. 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.\n\n## Induction\nThis group of definitions is [recursive](https://bliptext.com/articles/recursion), and requires some form of [mathematical induction](https://bliptext.com/articles/mathematical-induction) to define the universe of objects (forms and numbers) that occur in them.\nThe only surreal numbers reachable via finite induction are the [dyadic fractions](https://bliptext.com/articles/dyadic-rational); a wider universe is reachable given some form of [transfinite induction](https://bliptext.com/articles/transfinite-induction).\n\n### Induction rule\nS0 =\n, in which 0 consists of the single form\n.\nSn\nis the set of all surreal numbers that are generated by the construction rule from subsets of.\nThe base case is actually a special case of the induction rule, with 0 taken as a label for the \"least ordinal\".\nSince there exists no\nSi\nwith\ni < 0\n, the expression is the empty set; the only subset of the empty set is the empty set, and therefore\nS0\nconsists of a single surreal form lying in a single equivalence class 0.\nFor every finite ordinal number n,\nSn\nis [well-ordered](https://bliptext.com/articles/well-order) by the ordering induced by the comparison rule on the surreal numbers.\nThe first iteration of the induction rule produces the three numeric forms < < (the form is non-numeric because\n0 ≤ 0\n). 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](https://bliptext.com/articles/ring-mathematics); 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.\nFor every\ni < n\n, since every valid form in\nSi\nis also a valid form in\nSn\n, all of the numbers in\nSi\nalso appear in\nSn\n(as supersets of their representation in\nSi\n). (The set union expression appears in our construction rule, rather than the simpler form\nSn−1\n, so that the definition also makes sense when n is a [limit ordinal](https://bliptext.com/articles/limit-ordinal).) Numbers in\nSn\nthat are a superset of some number in\nSi\nare said to have been inherited from generation i. The smallest value of α for which a given surreal number appears in\nSα\nis called its birthday.\nFor example, the birthday of 0 is 0, and the birthday of −1 is 1.\nA second iteration of the construction rule yields the following ordering of equivalence classes:\n= = =\n< =\n< =\n< = = =\n< =\n< =\n< = = =\nComparison of these equivalence classes is consistent, irrespective of the choice of form.\nThree observations follow:\nS2\ncontains four new surreal numbers.\nTwo contain extremal forms: contains all numbers from previous generations in its right set, and contains all numbers from previous generations in its left set.\nThe others have a form that partitions all numbers from previous generations into two non-empty sets.\nThe 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.\nThe informal interpretations of and 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.\nThese labels will also be justified by the rules for surreal addition and multiplication below.\nThe 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).\nEither 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.\nWe retain the labels from the previous generation for these \"old\" numbers, and write the ordering above using the old and new labels:\nThe third observation extends to all surreal numbers with finite left and right sets.\n(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.\n\n### Birthday property\nA form\nx =\noccurring in generation n represents a number inherited from an earlier generation\ni < n\nif and only if there is some number in\nSi\nthat 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\nSn\nthat is a superset of the representation of c in generation i.\n\n# Arithmetic\nThe addition, negation (additive inverse), and multiplication of surreal number forms\nx =\nand\ny =\nare defined by three recursive formulas.\n\n## Negation\nNegation of a given number\nx =\nis defined by\nwhere the negation of a set S of numbers is given by the set of the negated elements of S:\nThis 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.\nThis only makes sense if the result is the same, irrespective of the choice of form of the operand.\nThis can be proved inductively using the fact that the numbers occurring in\nXL\nand\nXR\nare drawn from generations earlier than that in which the form x first occurs, and observing the special case:\n\n## Addition\nThe definition of addition is also a recursive formula:\nwhere\n.\nThis formula involves sums of one of the original operands and a surreal number drawn from the left or right set of the other.\nIt can be proved inductively with the special cases:\nFor example:\n1⁄2 + 1⁄2 = + =\n, which by the birthday property is a form of 1.\nThis justifies the label used in the previous section.\n\n### Subtraction\nSubtraction is defined with addition and negation:\n\n## Multiplication\nMultiplication can be defined recursively as well, beginning from the special cases involving 0, the [multiplicative identity](https://bliptext.com/articles/multiplicative-identity) 1, and its additive inverse −1:\nThe formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression that appears in the left set of the product of x and y. This is understood as, the set of numbers generated by picking all possible combinations of members of X_R and Y_R, and substituting them into the expression.\nFor example, to show that the square of 1⁄2 is 1⁄4:\n1⁄2 ⋅ 1⁄2 = ⋅ = = 1⁄4\n.\n\n## Division\nThe definition of division is done in terms of the reciprocal and multiplication:\nwhere\nfor positive y. Only positive\nyL\nare permitted in the formula, with any nonpositive terms being ignored (and\nyR\nare always positive). This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets of y, but also recursion in that the members of the left and right sets of\n1⁄y\nitself.\n0 is always a member of the left set of\n1⁄y\n, and that can be used to find more terms in a recursive fashion.\nFor example, if\n}\n, then we know a left term of 1⁄3 will be 0.\nThis in turn means\n1 + (2 − 3)0⁄2 = 1⁄2\nis a right term.\nThis means\nis a left term.\nThis means\nwill be a right term.\nContinuing, this gives\nFor negative y,\n1⁄y\nis given by\nIf\ny = 0\n, then\n1⁄y\nis undefined.\n\n## Consistency\nIt can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:\n0 =\nand multiplicative identity\n1 =\n.\nWith these rules one can now verify that the numbers found in the first few generations were properly labeled.\nThe construction rule is repeated to obtain more generations of surreals:\nS0 =\nS1 =\nS2 =\nS3 =\nS4 =\n\n## Arithmetic closure\nFor each [natural number](https://bliptext.com/articles/natural-number) (finite ordinal) n, all numbers generated in\nSn\nare [dyadic fractions](https://bliptext.com/articles/dyadic-fraction), i.e., can be written as an [irreducible fraction](https://bliptext.com/articles/irreducible-fraction)\na⁄2b\n, where a and b are [integers](https://bliptext.com/articles/integer) and\n0 ≤ b < n\n.\nThe set of all surreal numbers that are generated in some\nSn\nfor finite n may be denoted as.\nOne may form the three classes\nof which\nS∗\nis the union.\nNo individual\nSn\nis closed under addition and multiplication (except\nS0\n), but\nS∗\nis; it is the subring of the rationals consisting of all dyadic fractions.\nThere are infinite ordinal numbers β for which the set of surreal numbers with birthday less than β is closed under the different arithmetic operations.\nFor any ordinal α, the set of surreal numbers with birthday less than\nβ = ωα\n(using [powers of ω](https://bliptext.com/articles/)) is closed under addition and forms a group; for birthday less than ωω α it is closed under multiplication and forms a ring; and for birthday less than an (ordinal) [epsilon number](https://bliptext.com/articles/epsilon-number-mathematics) εα it is closed under multiplicative inverse and forms a field.\nThe latter sets are also closed under the exponential function as defined by Kruskal and Gonshor.\nHowever, 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](https://bliptext.com/articles/proper-class).\nWith their ordering and algebraic operations they constitute an [ordered field](https://bliptext.com/articles/ordered-field), with the caveat that they do not form a [set](https://bliptext.com/articles/set-mathematics).\nIn fact it is the biggest ordered field, in that every ordered field is a subfield of the surreal numbers.\nThe class of all surreal numbers is denoted by the symbol \\mathbb{No}.\n\n# Infinity\nDefine\nSω\nas the set of all surreal numbers generated by the construction rule from subsets of\nS∗\n. (This is the same inductive step as before, since the ordinal number ω 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 only be performed in a set theory that allows such a union.) A unique infinitely large positive number occurs in\nSω\nSω\nalso contains objects that can be identified as the [rational numbers](https://bliptext.com/articles/rational-number).\nFor example, the ω-complete form of the fraction 1⁄3 is given by:\nThe product of this form of 1⁄3 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.\nNot only do all the rest of the [rational numbers](https://bliptext.com/articles/rational-number) appear in\nSω\nThe only infinities in\nSω\nare ω and\n−ω\nSω\namong the reals.\nConsider the smallest positive number in\nSω\n. This number is larger than zero but less than all positive dyadic fractions.\nIt is therefore an [infinitesimal](https://bliptext.com/articles/infinitesimal) number, often labeled ε.\nThe ω-complete form of ε (respectively −ε) is the same as the ω-complete form of 0, except that 0 is included in the left (respectively right) set.\nThe only \"pure\" infinitesimals in\nSω\nare ε and its additive inverse −ε; adding them to any dyadic fraction y produces the numbers\ny ± ε\n, which also lie in\nSω\n.\nOne can determine the relationship between ω and ε by multiplying particular forms of them to obtain:\nω · ε =\n. This expression is only well-defined in a set theory which permits transfinite induction up to\nSω2\n. In such a system, one can demonstrate that all the elements of the left set of\nωSω·Sωε\nare positive infinitesimals and all the elements of the right set are positive infinities, and therefore\nωSω·Sωε\nis the oldest positive finite number, 1.\nConsequently,\n1⁄ε = ω\n. Some authors systematically use\nω−1\nin place of the symbol ε.\n\n## Contents of Sω\nGiven any\nx =\nin\nSω\n, exactly one of the following is true:\nx = 0\nn ≥ 0\nis greater than every element of L, in which case x equals the smallest such integer n;\n+ω\nn ≤ 0\nis less than every element of R, in which case x equals the largest such integer n;\n−ω\ny + ε\ny − ε\nSω\nis not an algebraic field, because it is not closed under arithmetic operations; consider ω+1, whose form\ndoes not lie in any number in\nSω\n. The maximal subset of\nSω\nthat is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities\n±ω\n, the infinitesimals\n±ε\n, and the infinitesimal neighbors\ny ± ε\nof each nonzero dyadic fraction y.\nThis construction of the real numbers differs from the [Dedekind cuts](https://bliptext.com/articles/dedekind-cut) of [standard analysis](https://bliptext.com/articles/real-analysis) in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in\nSω\nwith its forms in previous generations.\n(The ω-complete forms of real elements of\nSω\nare 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 Q of\nSω\ncontaining all elements x such that\nx b = a\nfor some a and some nonzero b, both drawn from\nS∗\n. By demonstrating that Q 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 Q is reachable from\nS∗\nby a finite series (no longer than two, actually) of arithmetic operations including multiplicative inversion, one can show that Q is strictly smaller than the subset of\nSω\nidentified with the reals.\nThe set\nSω\nhas the same [cardinality](https://bliptext.com/articles/cardinality) as the real numbers R. This can be demonstrated by exhibiting surjective mappings from\nSω\nto the closed unit interval I of R and vice versa.\nMapping\nSω\nonto I is routine; map numbers less than or equal to ε (including\n−ω\n) to 0, numbers greater than or equal to\n1 − ε\n(including ω) to 1, and numbers between ε and\n1 − ε\nto their equivalent in I (mapping the infinitesimal neighbors\ny±ε\nof each dyadic fraction y, along with y itself, to y). To map I onto\nSω\n, map the (open) central third (1⁄3, 2⁄3) of I onto\n= 0\n= 1\nS∗\n, monotonically.\nThe residue of I consists of the [Cantor set](https://bliptext.com/articles/cantor-set)\n2ω\n, 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\nin\nSω\n. This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday ω.\n\n# Transfinite induction\nContinuing to perform transfinite induction beyond\nSω\nproduces more ordinal numbers α, each represented as the largest surreal number having birthday α.\n(This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is\nω+1 =\n. There is another positive infinite number in generation\nω+1\nω − 1 =\n. The surreal number\nω − 1\nis not an ordinal; the ordinal ω is not the successor of any ordinal.\nThis is a surreal number with birthday ω+1, which is labeled\nω − 1\non the basis that it coincides with the sum of\nand\n−1 =\n. Similarly, there are two new infinitesimal numbers in generation\nω + 1\nε⁄2 = ε · 1⁄2 =\n.\nAt a later stage of transfinite induction, there is a number larger than\nω + k\nfor all natural numbers k:\nThis number may be labeled\nω + ω\nboth because its birthday is\nω + ω\n(the first ordinal number not reachable from ω by the successor operation) and because it coincides with the surreal sum of ω and ω; it may also be labeled 2ω because it coincides with the product of\nand\n. It is the second limit ordinal; reaching it from ω via the construction step requires a transfinite induction on\nThis involves an infinite union of infinite sets, which is a \"stronger\" set theoretic operation than the previous transfinite induction required.\nNote that the conventional addition and multiplication of ordinals does not always coincide with these operations on their surreal representations.\nThe sum of ordinals\n1 + ω\nequals ω, but the surreal sum is commutative and produces\n1 + ω = ω + 1 > ω\n. The addition and multiplication of the surreal numbers associated with ordinals coincides with the [natural sum and natural product](https://bliptext.com/articles/ordinal-arithmetic) of ordinals.\nJust as 2ω is bigger than\nω + n\nfor any natural number n, there is a surreal number ω⁄2 that is infinite but smaller than\nω − n\nfor any natural number n.\nThat is, ω⁄2 is defined by\nω⁄2 =\nwhere on the right hand side the notation x − Y is used to mean\n. It can be identified as the product of ω and the form of 1⁄2.\nThe birthday of ω⁄2 is the limit ordinal ω2.\n\n# Powers of ω and the Conway normal form\nTo classify the \"orders\" of infinite and infinitesimal surreal numbers, also known as [archimedean](https://bliptext.com/articles/archimedean-property) classes, Conway associated to each surreal number x the surreal number\nwhere r and s range over the positive real numbers.\nIf x < y then ωy is \"infinitely greater\" than ωx, in that it is greater than r ωx for all real numbers r.\nPowers of ω also satisfy the conditions\nso they behave the way one would expect powers to behave.\nEach power of ω 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.\nThus, for every positive surreal number x there will always exist some positive real number r and some surreal number y so that\nx − rωy\nis \"infinitely smaller\" than x.\nThe exponent y is the \"base ω logarithm\" of x, defined on the positive surreals; it can be demonstrated that logω maps the positive surreals onto the surreals and that\nThis gets extended by transfinite induction so that every surreal number has a \"normal form\" analogous to the [Cantor normal form](https://bliptext.com/articles/ordinal-arithmetic) for ordinal numbers.\nThis is the Conway normal form: Every surreal number x may be uniquely written as where every rα is a nonzero real number and the yαs form a strictly decreasing sequence of surreal numbers.\nThis \"sum\", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number.\n(Zero corresponds of course to the case of an empty sequence, and is the only surreal number with no leading exponent.)\nLooked at in this manner, the surreal numbers resemble a [power series field](https://bliptext.com/articles/formal-power-series), 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.\nThis is the basis for the formulation of the surreal numbers as a [Hahn series](https://bliptext.com/articles/).\n\n# Gaps and continuity\nIn 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.\nConway defines a gap as\nsuch that every element of L is less than every element of R, and ; this is not a number because at least one of the sides is a proper class.\nThough similar, gaps are not quite the same as [Dedekind cuts](https://bliptext.com/articles/dedekind-cut), but we can still talk about a completion of the surreal numbers with the natural ordering which is a (proper class-sized) [linear continuum](https://bliptext.com/articles/linear-continuum).\nFor instance there is no least positive infinite surreal, but the gap\nis greater than all real numbers and less than all positive infinite surreals, and is thus the least upper bound of the reals in.\nSimilarly the gap is larger than all surreal numbers.\n(This is an [esoteric pun](https://bliptext.com/articles/mathematical-joke): In the general construction of ordinals, α \"is\" the set of ordinals smaller than α, and we can use this equivalence to write in the surreals; \\mathbb{On} denotes the class of ordinal numbers, and because \\mathbb{On} is [cofinal](https://bliptext.com/articles/cofinal-mathematics) in \\mathbb{No} we have by extension.)\nWith a bit of set-theoretic care, \\mathbb{No} can be equipped with a topology where the [open sets](https://bliptext.com/articles/open-set) are unions of open intervals (indexed by proper sets) and continuous functions can be defined.\nAn equivalent of [Cauchy sequences](https://bliptext.com/articles/cauchy-sequence) 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\nwith aα decreasing and having no lower bound in \\mathbb{No}.\n(All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as ∞ and \\mathbb{On}).\n\n# Exponential function\nBased on unpublished work by [Kruskal](https://bliptext.com/articles/martin-david-kruskal), a construction (by transfinite induction) that extends the real [exponential function](https://bliptext.com/articles/exponential-function) exp(x) (with base e) to the surreals was carried through by Gonshor.\n\n## Other exponentials\nThe [powers of ω](https://bliptext.com/articles/) function is also an exponential function, but does not have the properties desired for an extension of the function on the reals.\nIt will, however, be needed in the development of the base-e exponential, and it is this function that is meant whenever the notation ωx is used in the following.\nWhen y is a dyadic fraction, the [power function](https://bliptext.com/articles/power-function) ,\nx ↦ xy\nmay be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively.\nIts values are completely determined by the basic relation\nxy+z = xy · xz\n, and where defined it necessarily agrees with any other [exponentiation](https://bliptext.com/articles/exponentiation) that can exist.\n\n## Basic induction\nThe induction steps for the surreal exponential are based on the series expansion for the real exponential,\nmore specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones.\nFor x positive these are denoted [x]n and include all partial sums; for x negative but finite, [x]2n+1 denotes the odd steps in the series starting from the first one with a positive real part (which always exists).\nFor x negative infinite the odd-numbered partial sums are strictly decreasing and the [x]2n+1 notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.\nThe relations that hold for real x < y are then\nand\nand this can be extended to the surreals with the definition\nThis is well-defined for all surreal arguments (the value exists and does not depend on the choice of zL and zR).\n\n## Results\nUsing this definition, the following hold:\nx < y ⇒ 0 < exp x < exp y\nexp(x+y) = exp x · exp y\nlog = exp–1\nexp 0 = 1, exp 1 = e\n)\nx – 1\nx = Σα<βrαωaα\nwhere all\naα > 0\n, then\nexp x = ωΣα<βrαω g(aα)\nx = ωΣα<βrαω bα\n, the inverse is given by\nlog x = Σα<βrαωg –1(bα)\n(ωΣγ<δtγω bγ )·r·(1 + Σα<βsαωaα)\n, for\naα < 0\n, where each factor has a form for which a way of calculating the logarithm has been given above; the sum is then the general logarithm\n\n## Examples\nThe surreal exponential is essentially given by its behaviour on positive powers of ω, i.e., the function g(a), combined with well-known behaviour on finite numbers.\nOnly examples of the former will be given.\nIn addition, g(a) = a 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 ω (ωω · · ω for some number of levels).\n\n## Exponentiation\nA general exponentiation can be defined as xy = exp(y · log x), giving an interpretation to expressions like 2ω = exp(ω · log 2) = ωlog 2 · ω.\nAgain it is essential to distinguish this definition from the \"powers of ω\" function, especially if ω may occur as the base.\n\n# Surcomplex numbers\nA **surcomplex number** is a number of the form\na + bi\n, where a and b are surreal numbers and\ni\nis the square root of\n−1\n. The surcomplex numbers form an [algebraically closed field](https://bliptext.com/articles/algebraically-closed-field) (except for being a [proper class](https://bliptext.com/articles/proper-class)), [isomorphic](https://bliptext.com/articles/isomorphic-mathematics) to the [algebraic closure](https://bliptext.com/articles/algebraic-closure) of the field generated by extending the [rational numbers](https://bliptext.com/articles/rational-numbers) by a [proper class](https://bliptext.com/articles/proper-class) of [algebraically independent](https://bliptext.com/articles/algebraically-independent) [transcendental](https://bliptext.com/articles/transcendental-mathematics) elements.\nUp to field [isomorphism](https://bliptext.com/articles/isomorphism), this fact characterizes the field of surcomplex numbers within any fixed set theory.\n\n# Games\nThe definition of surreal numbers contained one restriction: each element of L must be strictly less than each element of R. If this restriction is dropped we can generate a more general class known as **games**.\nAll games are constructed according to this rule:\nAddition, negation, and comparison are all defined the same way for both surreal numbers and games.\nEvery surreal number is a game, but not all games are surreal numbers, e.g. the game [star (game theory)](https://bliptext.com/articles/star-game-theory) is not a surreal number.\nThe class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers.\nThe class of surreal numbers forms a [field](https://bliptext.com/articles/field-mathematics), but the class of games does not.\nThe surreals have a [total order](https://bliptext.com/articles/total-order): given any two surreals, they are either equal, or one is greater than the other.\nThe games have only a [partial order](https://bliptext.com/articles/partial-order): there exist pairs of games that are neither equal, greater than, nor less than each other.\nEach surreal number is either positive, negative, or zero.\nEach game is either positive, negative, [zero](https://bliptext.com/articles/zero-game), or [fuzzy](https://bliptext.com/articles/fuzzy-game) (incomparable with [zero](https://bliptext.com/articles/zero-game), such as\n).\nA move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player.\nA player who cannot move because the choice is from the empty set has lost.\nA 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](https://bliptext.com/articles/fuzzy-game) for the first player to move.\nIf x, y, and z are surreals, and\nx = y\n, then\nx z = y z\n. However, if x, y, and z are games, and\nx = y\n, then it is not always true that\nx z = y z\n. Note that \"=\" here means equality, not identity.\n\n# Application to combinatorial game theory\nThe surreal numbers were originally motivated by studies of the game [Go](https://bliptext.com/articles/go-board-game), and there are numerous connections between popular games and the surreals.\nIn this section, we will use a capitalized Game for the mathematical object\n, and the lowercase game for recreational games like [Chess](https://bliptext.com/articles/chess) or [Go](https://bliptext.com/articles/go-board-game).\nWe consider games with these properties:\nFor most games, the initial board position gives no great advantage to either player.\nAs the game progresses and one player starts to win, board positions will occur in which that player has a clear advantage.\nFor analyzing games, it is useful to associate a Game with every board position.\nThe value of a given position will be the Game {L|R}, where L is the set of values of all the positions that can be reached in a single move by Left.\nSimilarly, R is the set of values of all the positions that can be reached in a single move by Right.\nThe zero Game (called 0) is the Game where L and R are both empty, so the player to move next (L or R) immediately loses.\nThe sum of two Games G = and H = is defined as the Game G + H = 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.\nOne can imagine two chess boards between two players, with players making moves alternately, but with complete freedom as to which board to play on.\nIf G is the Game {L | R}, −G is the Game {−R | −L}, i.e. with the role of the two players reversed.\nIt is easy to show G – G = 0 for all Games G (where G – H is defined as G + (–H)).\nThis simple way to associate Games with games yields a very interesting result.\nSuppose two perfect players play a game starting with a given position whose associated Game is x.\nWe can classify all Games into four classes as follows:\nMore generally, we can define G > H as G – H > 0, and similarly for <, = and ||.\nThe notation G || H means that G and H are incomparable.\nG || H is equivalent to G − H || 0, i.e. that G > H, G < H and G = H are all false.\nIncomparable 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.\nA game confused with zero is said to be [fuzzy](https://bliptext.com/articles/fuzzy-game), as opposed to [positive, negative, or zero](https://bliptext.com/articles/sign-mathematics).\nAn example of a fuzzy game is [star (*)](https://bliptext.com/articles/star-game-theory).\nSometimes 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.\nFor 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.\nEach island is like a separate game of Go, played on a very small board.\nIt would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game.\nThis doesn't appear to be easy to do.\nFor 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.\nFortunately, there is a way to do this analysis.\nThe following theorem can be applied:\nx + y\n.\nA game composed of smaller games is called the [disjunctive sum](https://bliptext.com/articles/disjunctive-sum) of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the [disjunctive sum](https://bliptext.com/articles/disjunctive-sum) of the addends.\nHistorically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here.\nHe was analyzing [Go endgames](https://bliptext.com/articles/go-terms), 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](https://bliptext.com/articles/disjunctive-sum).\nFrom this he invented the concept of a Game and the addition operator for it.\nFrom there he moved on to developing a definition of negation and comparison.\nThen he noticed that a certain class of Games had interesting properties; this class became the surreal numbers.\nFinally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.\n\n# Alternative realizations\nAlternative approaches to the surreal numbers complement Conway's exposition in terms of games.\n\n## Sign expansion\n\n### Definitions\nIn what is now called the sign-expansion or sign-sequence of a surreal number, a surreal number is a [function](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) is an [ordinal](https://bliptext.com/articles/ordinal-number) and whose co[domain](https://bliptext.com/articles/domain-of-a-function) is.\nThis is equivalent to Conway's L-R sequences.\nDefine the binary predicate \"simpler than\" on numbers by x is simpler than y if x is a [proper subset](https://bliptext.com/articles/subset) of y, i.e. if dom(x) < dom(y) and x(α) = y(α) for all α < dom(x).\nFor surreal numbers define the binary relation < to be lexicographic order (with the convention that \"undefined values\" are greater than −1 and less than 1).\nSo x < y if one of the following holds:\nEquivalently, let δ(x,y) = min({ dom(x), dom(y)} ∪ { α : α < dom(x) ∧ α < dom(y) ∧ x(α) ≠ y(α) }),\nso that x = y if and only if δ(x,y) = dom(x) = dom(y).\nThen, for numbers x and y, x < y if and only if one of the following holds:\nFor numbers x and y, x ≤ y if and only if x < y ∨ x = y, and x > y if and only if y < x.\nAlso x ≥ y if and only if y ≤ x.\nThe relation < is [transitive](https://bliptext.com/articles/transitive-relation), and for all numbers x and y, exactly one of x < y, x = y, x > y, holds (law of [trichotomy](https://bliptext.com/articles/trichotomy-mathematics)).\nThis means that < is a [linear order](https://bliptext.com/articles/linear-order) (except that < is a proper class).\nFor sets of numbers, L and R such that ∀x ∈ L ∀y ∈ R (x < y), there exists a unique number z such that\nFurthermore, z is constructible from L and R by transfinite induction.\nz is the simplest number between L and R.\nLet the unique number z be denoted by σ(L,R).\nFor a number x, define its left set L(x) and right set R(x) by\nthen σ(L(x),R(x)) = x.\nOne advantage of this alternative realization is that equality is identity, not an inductively defined relation.\nUnlike Conway's 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.\nHowever, similar definitions can be made that eliminate the need for prior construction of the ordinals.\nFor 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 ∀g ∈ dom f (∀h ∈ dom g (h ∈ dom f )) and whose range is.\n\"Simpler than\" is very simply defined now—x is simpler than y if x ∈ dom y.\nThe total ordering is defined by considering x and y as sets of ordered pairs (as a function is normally defined): Either x = y, or else the surreal number z = x ∩ y is in the domain of x or the domain of y (or both, but in this case the signs must disagree).\nWe then have x < y if x(z) = − or y(z) = + (or both).\nConverting these functions into sign sequences is a straightforward task; arrange the elements of dom f in order of simplicity (i.e., inclusion), and then write down the signs that f assigns to each of these elements in order.\nThe ordinals then occur naturally as those surreal numbers whose range is.\n\n### Addition and multiplication\nThe sum x + y of two numbers, x and y, is defined by induction on dom(x) and dom(y) by x + y = σ(L,R), where\nL = ∪\n,\nR = ∪\n.\nThe additive identity is given by the number 0 = { }, i.e. the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number x is the number −x, given by dom(−x) = dom(x), and, for α < dom(x), (−x)(α) = −1 if x(α) = +1, and (−x)(α) = +1 if x(α) = −1.\nIt follows that a number x is [positive](https://bliptext.com/articles/positive-number) if and only if 0 < dom(x) and x(0) = +1, and x is [negative](https://bliptext.com/articles/negative-number) if and only if 0 < dom(x) and x(0) = −1.\nThe product xy of two numbers, x and y, is defined by induction on dom(x) and dom(y) by xy = σ(L,R), where\nL = ∪\n,\nR = ∪\n.\nThe multiplicative identity is given by the number 1 =, i.e. the number 1 has domain equal to the ordinal 1, and 1(0) = +1.\n\n### Correspondence with Conway's realization\nThe map from Conway's realization to sign expansions is given by f = σ(M,S), where M = and S =.\nThe inverse map from the alternative realization to Conway's realization is given by g(x) =, where L = and R =.\n\n## Axiomatic approach\nIn another approach to the surreals, given by Alling, explicit construction is bypassed altogether.\nInstead, a set of axioms is given that any particular approach to the surreals must satisfy.\nMuch like the [axiomatic approach](https://bliptext.com/articles/real-numbers) to the reals, these axioms guarantee uniqueness up to isomorphism.\nA triple is a surreal number system if and only if the following hold:\nx ∈ A\nand\ny ∈ B\n, x < y (using Alling's terminology, 〈 A, B 〉 is a \"Conway cut\" of \\mathbb{No}).\nThen there exists a unique such that b(z) is minimal and for all\nx ∈ A\nand all\ny ∈ B\n, x < z < y.\n(This axiom is often referred to as \"Conway's Simplicity Theorem\".)\nBoth Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.\nGiven these axioms, Alling derives Conway's original definition of ≤ and develops surreal arithmetic.\n\n## Simplicity hierarchy\nA construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich.\nThe 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.\nThis construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation.\n\n## Hahn series\nAlling also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of [Hahn series](https://bliptext.com/articles/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).\nThis provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.\nThis 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.,\nν(ω) = −1\n. The valuation ring then consists of the finite surreal numbers (numbers with a real and/or an infinitesimal part).\nThe 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.\n\n# Relation to hyperreals\nPhilip Ehrlich has constructed an isomorphism between Conway's maximal surreal number field and the maximal [hyperreals](https://bliptext.com/articles/hyperreal-field) in [von Neumann–Bernays–Gödel set theory](https://bliptext.com/articles/von-neumann-bernays-g-del-set-theory).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:36:34.309Z","updated_at":"2025-01-25T17:36:34.309Z","search_vector":"'number':2 'surreal':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **surreal number** system is a [totally ordered](https://bliptext.com/articles/total-order) [proper class](https://bliptext.com/articles/proper-class) containing not only the [real numbers](https://bliptext.com/articles/real-number) but also [infinite](https://bliptext.com/articles/infinity) and [infinite](https://bliptext.com/articles/infinity)simal numbers, respectively larger or smaller in [absolute value](https://bliptext.com/articles/absolute-value) than any positive real number.\nResearch on the [Go endgame](https://bliptext.com/articles/go-endgame) by [John Horton Conway](https://bliptext.com/articles/john-horton-conway) led to the original definition and construction of surreal numbers.\nConway's construction was introduced in [Donald Knuth](https://bliptext.com/articles/donald-knuth)'s 1974 book **Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness**.\nThe surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an [ordered field](https://bliptext.com/articles/ordered-field).\nIf formulated in [von Neumann–Bernays–Gödel set theory](https://bliptext.com/articles/von-neumann-bernays-g-del-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](https://bliptext.com/articles/rational-function), the [Levi-Civita field](https://bliptext.com/articles/levi-civita-field), the [superreal numbers](https://bliptext.com/articles/superreal-number) (including the [hyperreal numbers](https://bliptext.com/articles/hyperreal-number)) can be realized as subfields of the surreals.\nThe surreals also contain all [transfinite](https://bliptext.com/articles/transfinite-number) [ordinal numbers](https://bliptext.com/articles/ordinal-number); the arithmetic on them is given by the [natural operations](https://bliptext.com/articles/ordinal-arithmetic).\nIt has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is [isomorphic](https://bliptext.com/articles/isomorphic) to the maximal class surreal field.\n# [A visualization of the surreal number tree. | upload.wikimedia.org/wikipedia/commons/4/49/Surreal///number///tree.svg]\n\n# History of the concept\nResearch on the [Go endgame](https://bliptext.com/articles/go-strategy-and-tactics) by [John Horton Conway](https://bliptext.com/articles/john-horton-conway) led to the original definition and construction of the surreal numbers.\nConway's construction was introduced in [Donald Knuth](https://bliptext.com/articles/donald-knuth)'s 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.\nIn his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers.\nConway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book [On Numbers and Games](https://bliptext.com/articles/on-numbers-and-games).\nA separate route to defining the surreals began in 1907, when [Hans Hahn](https://bliptext.com/articles/hans-hahn-mathematician) introduced [Hahn series](https://bliptext.com/articles/hahn-series) as a generalization of [formal power series](https://bliptext.com/articles/formal-power-series), and [Felix Hausdorff](https://bliptext.com/articles/felix-hausdorff) introduced certain ordered sets called [[η set|\nηα\n-sets]] for ordinals α and asked if it was possible to find a compatible ordered group or field structure.\nIn 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.\nIf the surreals are considered as 'just' a proper-class-sized real closed field, Alling's 1962 paper handles the case of [strongly inaccessible](https://bliptext.com/articles/inaccessible-cardinal) cardinals which can naturally be considered as proper classes by cutting off the [cumulative hierarchy of the universe](https://bliptext.com/articles/von-neumann-universe) one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense.\nThere 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.\nThis 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.\n\n# Description\n\n## Notation\nIn the context of surreal numbers, an [ordered pair](https://bliptext.com/articles/ordered-pair) of sets L and R, which is written as\n(L, R)\nin many other mathematical contexts, is instead written\nincluding the extra space adjacent to each brace.\nWhen a set is empty, it is often simply omitted.\nWhen a set is explicitly described by its elements, the pair of braces that encloses the list of elements is often omitted.\nWhen a union of sets is taken, the operator that represents that is often a comma.\nFor example, instead of\n(L1 ∪ L2 ∪, ∅)\n, which is common notation in other contexts, we typically write\n.\n\n## Outline of construction\nIn the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers a and b,\na ≤ b\nor\nb ≤ a\n. (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\nrepresents a number intermediate in value between all the members of L and all the members of R.\nDifferent subsets may end up defining the same number:\nand\nmay define the same number even if\nL ≠ L′\nand\nR ≠ R′\n. (A similar phenomenon occurs when [rational numbers](https://bliptext.com/articles/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](https://bliptext.com/articles/equivalence-class) of representations of the form\nthat designate the same number.\nIn the first stage of construction, there are no previously existing numbers so the only representation must use the empty set:\n. This representation, where L and R are both empty, is called 0.\nSubsequent stages yield forms like\n= 1\n= 2\n= 3\nand\n= −1\n= −2\n= −3\nThe integers are thus contained within the surreal numbers.\n(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).\nSimilarly, representations such as\n= 1⁄2\n= 1⁄4\n= 3⁄4\narise, so that the [dyadic rationals](https://bliptext.com/articles/dyadic-rational) (rational numbers whose denominators are powers of 2) are contained within the surreal numbers.\nAfter an infinite number of stages, infinite subsets become available, so that any [real number](https://bliptext.com/articles/real-number) a can be represented by\n,\nwhere\nLa\nis the set of all dyadic rationals less than a and\nRa\nis the set of all dyadic rationals greater than a (reminiscent of a [Dedekind cut](https://bliptext.com/articles/dedekind-cut)).\nThus the real numbers are also embedded within the surreals.\nThere are also representations like\n= ω\n= ε\nwhere ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number.\nMoreover, 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\n2ω\nor\nω − 1\nand so forth.\n\n# Construction\nSurreal numbers are [constructed inductively](https://bliptext.com/articles/inductive-definition) as [equivalence classes](https://bliptext.com/articles/equivalence-class) of [pairs](https://bliptext.com/articles/ordered-pair) 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.\nThe construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.\n\n## Forms\nA form is a pair of sets of surreal numbers, called its left set and its right set.\nA form with left set L and right set R is written\n. When L and R are given as lists of elements, the braces around them are omitted.\nEither or both of the left and right set of a form may be the empty set.\nThe form\nwith both left and right set empty is also written\n.\n\n## Numeric forms and their equivalence classes\n**Construction rule**\nis 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](https://bliptext.com/articles/order-theory) ≤ given by the comparison rule below.\nThe numeric forms are placed in equivalence classes; each such equivalence class is a surreal number.\nThe 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).\n**Equivalence rule**\nx ≤ y\nand\ny ≤ x\n.\nAn [ordering relationship](https://bliptext.com/articles/order-theory) must be [antisymmetric](https://bliptext.com/articles/antisymmetric-relation), i.e., it must have the property that\nx = y\n(i. e.,\nx ≤ y\nand\ny ≤ x\nare 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).\nThe equivalence class containing\nis labeled 0; in other words,\nis a form of the surreal number 0.\n\n## Order\nThe recursive definition of surreal numbers is completed by defining comparison:\nGiven numeric forms\nx =\nand\ny =\n,\nx ≤ y\nif and only if both:\nxL ∈ XL\nsuch that\ny ≤ xL\n. That is, every element in the left part of x is strictly smaller than y.\nyR ∈ YR\nsuch that\nyR ≤ x\n. 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.\n\n## Induction\nThis group of definitions is [recursive](https://bliptext.com/articles/recursion), and requires some form of [mathematical induction](https://bliptext.com/articles/mathematical-induction) to define the universe of objects (forms and numbers) that occur in them.\nThe only surreal numbers reachable via finite induction are the [dyadic fractions](https://bliptext.com/articles/dyadic-rational); a wider universe is reachable given some form of [transfinite induction](https://bliptext.com/articles/transfinite-induction).\n\n### Induction rule\nS0 =\n, in which 0 consists of the single form\n.\nSn\nis the set of all surreal numbers that are generated by the construction rule from subsets of.\nThe base case is actually a special case of the induction rule, with 0 taken as a label for the \"least ordinal\".\nSince there exists no\nSi\nwith\ni ω\n. The addition and multiplication of the surreal numbers associated with ordinals coincides with the [natural sum and natural product](https://bliptext.com/articles/ordinal-arithmetic) of ordinals.\nJust as 2ω is bigger than\nω + n\nfor any natural number n, there is a surreal number ω⁄2 that is infinite but smaller than\nω − n\nfor any natural number n.\nThat is, ω⁄2 is defined by\nω⁄2 =\nwhere on the right hand side the notation x − Y is used to mean\n. It can be identified as the product of ω and the form of 1⁄2.\nThe birthday of ω⁄2 is the limit ordinal ω2.\n\n# Powers of ω and the Conway normal form\nTo classify the \"orders\" of infinite and infinitesimal surreal numbers, also known as [archimedean](https://bliptext.com/articles/archimedean-property) classes, Conway associated to each surreal number x the surreal number\nwhere r and s range over the positive real numbers.\nIf x 0\n, then\nexp x = ωΣα H as G – H > 0, and similarly for H, G < H and G = H are all false.\nIncomparable 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.\nA game confused with zero is said to be [fuzzy](https://bliptext.com/articles/fuzzy-game), as opposed to [positive, negative, or zero](https://bliptext.com/articles/sign-mathematics).\nAn example of a fuzzy game is [star (*)](https://bliptext.com/articles/star-game-theory).\nSometimes 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.\nFor 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.\nEach island is like a separate game of Go, played on a very small board.\nIt would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game.\nThis doesn't appear to be easy to do.\nFor 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.\nFortunately, there is a way to do this analysis.\nThe following theorem can be applied:\nx + y\n.\nA game composed of smaller games is called the [disjunctive sum](https://bliptext.com/articles/disjunctive-sum) of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the [disjunctive sum](https://bliptext.com/articles/disjunctive-sum) of the addends.\nHistorically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here.\nHe was analyzing [Go endgames](https://bliptext.com/articles/go-terms), 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](https://bliptext.com/articles/disjunctive-sum).\nFrom this he invented the concept of a Game and the addition operator for it.\nFrom there he moved on to developing a definition of negation and comparison.\nThen he noticed that a certain class of Games had interesting properties; this class became the surreal numbers.\nFinally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.\n\n# Alternative realizations\nAlternative approaches to the surreal numbers complement Conway's exposition in terms of games.\n\n## Sign expansion\n\n### Definitions\nIn what is now called the sign-expansion or sign-sequence of a surreal number, a surreal number is a [function](https://bliptext.com/articles/function-mathematics) whose [domain](https://bliptext.com/articles/domain-of-a-function) is an [ordinal](https://bliptext.com/articles/ordinal-number) and whose co[domain](https://bliptext.com/articles/domain-of-a-function) is.\nThis is equivalent to Conway's L-R sequences.\nDefine the binary predicate \"simpler than\" on numbers by x is simpler than y if x is a [proper subset](https://bliptext.com/articles/subset) of y, i.e. if dom(x) < dom(y) and x(α) = y(α) for all α < dom(x).\nFor surreal numbers define the binary relation < to be lexicographic order (with the convention that \"undefined values\" are greater than −1 and less than 1).\nSo x < y if one of the following holds:\nEquivalently, let δ(x,y) = min({ dom(x), dom(y)} ∪ { α : α < dom(x) ∧ α < dom(y) ∧ x(α) ≠ y(α) }),\nso that x = y if and only if δ(x,y) = dom(x) = dom(y).\nThen, for numbers x and y, x < y if and only if one of the following holds:\nFor numbers x and y, x ≤ y if and only if x < y ∨ x = y, and x > y if and only if y < x.\nAlso x ≥ y if and only if y ≤ x.\nThe relation < is [transitive](https://bliptext.com/articles/transitive-relation), and for all numbers x and y, exactly one of x < y, x = y, x > y, holds (law of [trichotomy](https://bliptext.com/articles/trichotomy-mathematics)).\nThis means that < is a [linear order](https://bliptext.com/articles/linear-order) (except that < is a proper class).\nFor sets of numbers, L and R such that ∀x ∈ L ∀y ∈ R (x < y), there exists a unique number z such that\nFurthermore, z is constructible from L and R by transfinite induction.\nz is the simplest number between L and R.\nLet the unique number z be denoted by σ(L,R).\nFor a number x, define its left set L(x) and right set R(x) by\nthen σ(L(x),R(x)) = x.\nOne advantage of this alternative realization is that equality is identity, not an inductively defined relation.\nUnlike Conway's 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.\nHowever, similar definitions can be made that eliminate the need for prior construction of the ordinals.\nFor 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 ∀g ∈ dom f (∀h ∈ dom g (h ∈ dom f )) and whose range is.\n\"Simpler than\" is very simply defined now—x is simpler than y if x ∈ dom y.\nThe total ordering is defined by considering x and y as sets of ordered pairs (as a function is normally defined): Either x = y, or else the surreal number z = x ∩ y is in the domain of x or the domain of y (or both, but in this case the signs must disagree).\nWe then have x < y if x(z) = − or y(z) = + (or both).\nConverting these functions into sign sequences is a straightforward task; arrange the elements of dom f in order of simplicity (i.e., inclusion), and then write down the signs that f assigns to each of these elements in order.\nThe ordinals then occur naturally as those surreal numbers whose range is.\n\n### Addition and multiplication\nThe sum x + y of two numbers, x and y, is defined by induction on dom(x) and dom(y) by x + y = σ(L,R), where\nL = ∪\n,\nR = ∪\n.\nThe additive identity is given by the number 0 = { }, i.e. the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number x is the number −x, given by dom(−x) = dom(x), and, for α < dom(x), (−x)(α) = −1 if x(α) = +1, and (−x)(α) = +1 if x(α) = −1.\nIt follows that a number x is [positive](https://bliptext.com/articles/positive-number) if and only if 0 < dom(x) and x(0) = +1, and x is [negative](https://bliptext.com/articles/negative-number) if and only if 0 < dom(x) and x(0) = −1.\nThe product xy of two numbers, x and y, is defined by induction on dom(x) and dom(y) by xy = σ(L,R), where\nL = ∪\n,\nR = ∪\n.\nThe multiplicative identity is given by the number 1 =, i.e. the number 1 has domain equal to the ordinal 1, and 1(0) = +1.\n\n### Correspondence with Conway's realization\nThe map from Conway's realization to sign expansions is given by f = σ(M,S), where M = and S =.\nThe inverse map from the alternative realization to Conway's realization is given by g(x) =, where L = and R =.\n\n## Axiomatic approach\nIn another approach to the surreals, given by Alling, explicit construction is bypassed altogether.\nInstead, a set of axioms is given that any particular approach to the surreals must satisfy.\nMuch like the [axiomatic approach](https://bliptext.com/articles/real-numbers) to the reals, these axioms guarantee uniqueness up to isomorphism.\nA triple is a surreal number system if and only if the following hold:\nx ∈ A\nand\ny ∈ B\n, x < y (using Alling's terminology, 〈 A, B 〉 is a \"Conway cut\" of \\mathbb{No}).\nThen there exists a unique such that b(z) is minimal and for all\nx ∈ A\nand all\ny ∈ B\n, x < z < y.\n(This axiom is often referred to as \"Conway's Simplicity Theorem\".)\nBoth Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.\nGiven these axioms, Alling derives Conway's original definition of ≤ and develops surreal arithmetic.\n\n## Simplicity hierarchy\nA construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich.\nThe 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.\nThis construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation.\n\n## Hahn series\nAlling also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of [Hahn series](https://bliptext.com/articles/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).\nThis provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.\nThis 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.,\nν(ω) = −1\n. The valuation ring then consists of the finite surreal numbers (numbers with a real and/or an infinitesimal part).\nThe 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.\n\n# Relation to hyperreals\nPhilip Ehrlich has constructed an isomorphism between Conway's maximal surreal number field and the maximal [hyperreals](https://bliptext.com/articles/hyperreal-field) in [von Neumann–Bernays–Gödel set theory](https://bliptext.com/articles/von-neumann-bernays-g-del-set-theory).","revisionCount":0},"hasMore":false,"nextCursor":null}