12469:
10541:
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:
20:
4248:
14278:
10582:. 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.
10494:
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 G is the Game {L | R}, −G is the Game {−R | −L}, i.e. with the role of the two players reversed. It is easy to show G – G = 0 for all Games G (where G – H is defined as G + (–H)).
3908:
14382:
210:
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.
5700:. (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
3485:
2145:, which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form. This only makes sense if the result is the same, irrespective of the choice of form of the operand. This can be proved inductively using the fact that the numbers occurring in
12441:. 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.
4243:{\displaystyle {\frac {1}{y}}=\left\{\left.0,{\frac {1+(y_{R}-y)\left({\frac {1}{y}}\right)_{L}}{y_{R}}},{\frac {1+\left(y_{L}-y\right)\left({\frac {1}{y}}\right)_{R}}{y_{L}}}\,\,\right|\,\,{\frac {1+(y_{L}-y)\left({\frac {1}{y}}\right)_{L}}{y_{L}}},{\frac {1+(y_{R}-y)\left({\frac {1}{y}}\right)_{R}}{y_{R}}}\right\}}
6330:
3157:
259:
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
12413:
A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip
Ehrlich. 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
10493:
The 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. The sum of two Games G = { L1 | R1 } and H = { L2 | R2 } is defined as the Game G + H = { L1 + H, G + L2 | R1 + H, G + R2 } where the player to move chooses which of the Games to
7128:
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
538:
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
11331:
One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike 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
10540:
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
1660:
The informal interpretations of { 1 | } and { | −1 } are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of { 0 | 1 } and { −1 | 0 } are "the number halfway between 0 and 1" and "the number halfway
785:
is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the
209:
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
1712:
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
10489:
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
1753:
The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number { 1, 2 | 5, 8 } is therefore equivalent to { 2 | 5 }; one can
12426:
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.
10328:
A 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. A player who cannot move because the choice is from the empty set has lost. A positive game
8283:
s 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.)
1630:
contains four new surreal numbers. Two contain extremal forms: { | −1, 0, 1 } contains all numbers from previous generations in its right set, and { −1, 0, 1 | } contains all numbers from previous generations in its left set. The others have a form that partitions all numbers from previous
6103:
4648:
5560:
11335:
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
820:
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.
9445:
6154:
10490:
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. Similarly, R is the set of values of all the positions that can be reached in a single move by Right.
9860:
While there is no general inductive definition of log (unlike for exp), the partial results are given in terms of such definitions. In this way, the logarithm can be calculated explicitly, without reference to the fact that it's the inverse of the
3141:
193:
and asked if it was possible to find a compatible ordered group or field structure. In 1962, Norman Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α and, in 1987, he showed that taking
2415:
7043:
3684:
8495:
5976:
12514:
is not precisely correct; where this distinction is important, some authors use Field or FIELD to refer to a proper class that has the arithmetic properties of a field. One can obtain a true field by limiting the construction to a
4538:
2536:
12414:
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.
2877:
2748:
8835:
5605:
There are infinite ordinal numbers β for which the set of surreal numbers with birthday less than β is closed under the different arithmetic operations. For any ordinal α, the set of surreal numbers with birthday less than
5933:
5634:
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
4533:
4455:
3480:{\displaystyle {\begin{aligned}xy&=\{X_{L}\mid X_{R}\}\{Y_{L}\mid Y_{R}\}\\&=\left\{X_{L}y+xY_{L}-X_{L}Y_{L},X_{R}y+xY_{R}-X_{R}Y_{R}\mid X_{L}y+xY_{R}-X_{L}Y_{R},xY_{L}+X_{R}y-X_{R}Y_{L}\right\}\\\end{aligned}}}
8734:
8291:, 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
5404:
9703:
5797:
12043:
In another approach to the surreals, given by 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
10296:
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
2619:
2072:
12094:
10249:
The 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
4730:
9080:
5951:
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.
9183:
7855:
3899:
10309:: 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,
8578:
5409:
3162:
1438:). The equivalence class containing { 0 | } is labeled 1 and the equivalence class containing { | 0 } is labeled −1. These three labels have a special significance in the axioms that define a
14580:
4795:
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;
2233:
10521:
The notation G || H means that G and H are incomparable. G || H is equivalent to G − H || 0, i.e. that G > H, G < H and G = H are all false. Incomparable games are sometimes said to be
9583:
is given in Conway normal form, the set of exponents in the result is well-ordered and the coefficients are finite sums, directly giving the normal form of the result (which has a leading 1)
2964:
8930:
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-
8303:
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 a gap as
4841:
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:
2245:
8531:
8396:
5399:
8772:
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
6911:
1442:; 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.
8166:
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
3554:
2541:
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:
8410:
1398:
1333:
10497:
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
8989:
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
8358:
12430:
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.,
2133:
12317:
8977:
13775:
9550:
14090:
14013:
13974:
13936:
13908:
13880:
13852:
13740:
13707:
13679:
13651:
10677:
For surreal numbers define the binary relation < to be lexicographic order (with the convention that "undefined values" are greater than −1 and less than 1). So
79:
The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an
12286:
12225:
12189:
12157:
12127:
9809:
The normal form can be written out by multiplying the infinite part (a single power of ω) and the real exponential into the power series resulting from the infinitesimal
9806:
Any surreal number can be written as the sum of a pure infinite, a real and an infinitesimal part, and the exponential is the product of the partial results given above
8892:
8867:
8762:
8665:
8636:
8611:
5670:
5647:. In fact it is the biggest ordered field, in that every ordered field is a subfield of the surreal numbers. The class of all surreal numbers is denoted by the symbol
2752:
2623:
8775:
10025:
6325:{\displaystyle \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\}}
5832:
4460:
4382:
12540:
The set of dyadic fractions constitutes the simplest non-trivial group and ring of this kind; it consists of the surreal numbers with birthday less than ω = ω = ω.
9986:
5623:) is closed under addition and forms a group; for birthday less than ω it is closed under multiplication and forms a ring; and for birthday less than an (ordinal)
6855:
6792:
10029:
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
3738:
3711:
5631:
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.
1220:
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.
13080:
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,
5714:
4784:
Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday
7442:, 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
1434:
The first iteration of the induction rule produces the three numeric forms { | 0 } < { | } < { 0 | } (the form { 0 | 0 } is non-numeric because
2544:
2422:
1978:
12982:
12827:
7182:
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
6098:{\displaystyle \pi =\left\{3,{\tfrac {25}{8}},{\tfrac {201}{64}},\ldots \mid 4,{\tfrac {7}{2}},{\tfrac {13}{4}},{\tfrac {51}{16}},\ldots \right\}.}
10571:
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.
10525:
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
7857:
This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.
4679:
340:
are equivalent and denote the same number.) Each number is formed from an ordered pair of subsets of numbers already constructed: given subsets
8670:
12507:
12454:
84:
14585:
9018:
7864:
addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals
4773:
Addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday
9647:
4643:{\displaystyle {\frac {1}{3}}=\left\{\left.0,{\frac {1}{4}},{\frac {5}{16}},\ldots \,\right|\,{\frac {1}{2}},{\frac {3}{8}},\ldots \right\}}
4278:
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
7811:
3858:
304:
In the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers
1638:
that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers
12741:
9857:, 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
2174:
1339:
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
10574:
Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing
8869:. (All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as ∞ and
1708:(each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains
10578:, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their
3567:
87:, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the
13222:
14314:
13601:
7067:
that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities
5555:{\displaystyle {\begin{aligned}S_{0}&=\{0\}\\S_{+}&=\{x\in S_{*}:x>0\}\\S_{-}&=\{x\in S_{*}:x<0\}\end{aligned}}}
13317:
12054:
8499:
is greater than all real numbers and less than all positive infinite surreals, and is thus the least upper bound of the reals in
478:
In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set:
12986:
10162:. Again it is essential to distinguish this definition from the "powers of ω" function, especially if ω may occur as the base.
8584:: In the general construction of ordinals, α "is" the set of ordinals smaller than α, and we can use this equivalence to write
1656:
The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set.
11444:
assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is { + }.
7496:. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is
198:
to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.
10329:
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
13790:
9946:
constitute a field that is closed under exponentials, and is likewise an elementary extension of the real exponential field
8360:; 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
4825:
These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a
13307:
10424:, and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized
9082:
more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For
3487:
The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression
201:
If the surreals are considered as 'just' a proper-class-sized real closed field, Alling's 1962 paper handles the case of
13785:
13065:
An update of the classic 1976 book defining the surreal numbers, and exploring their connections to games: John Conway,
13059:
11436:) = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom
8536:
6356:-complete form of 0, except that 0 is included in the left (respectively right) set. The only "pure" infinitesimals in
14611:
13267:
7106:
in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in
4788:
will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than
2085:
1618:
Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow:
12583:
Even the most trivial-looking of these equalities may involve transfinite induction and constitute a separate theorem.
14528:
14421:
14365:
14352:
13512:
13181:
13173:
13154:
13139:
13107:
13089:
13074:
13055:
12963:
12893:
12801:
12617:
9104:
denotes the odd steps in the series starting from the first one with a positive real part (which always exists). For
13745:
9116:
notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.
1236:
to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable via
14411:
14163:
13368:
9440:{\displaystyle \exp z=\{0,\exp z_{L}\cdot _{n},\exp z_{R}\cdot _{2n+1}\mid \exp z_{R}/_{n},\exp z_{L}/_{2n+1}\}.}
4769:
It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:
134:
led to the original definition and construction of the surreal numbers. Conway's construction was introduced in
14241:
9990:, combined with well-known behaviour on finite numbers. Only examples of the former will be given. In addition,
8768:
are unions of open intervals (indexed by proper sets) and continuous functions can be defined. An equivalent of
12520:
8502:
8367:
938:. The elements of the left and right sets of a form are drawn from the universe of the surreal numbers (not of
8236:
This gets extended by transfinite induction so that every surreal number has a "normal form" analogous to the
5348:
14636:
14416:
14124:
12607:
12574:
Importantly, there is no claim that the collection of Cauchy sequences constitutes a class in NBG set theory.
12528:
114:. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is
13160:
A detailed philosophical development of the concept of surreal numbers as a most general concept of number:
12449:
Philip
Ehrlich has constructed an isomorphism between Conway's maximal surreal number field and the maximal
9015:
The induction steps for the surreal exponential are based on the series expansion for the real exponential,
66:
led to the original definition and construction of surreal numbers. Conway's construction was introduced in
14493:
14307:
13594:
13414:
13233:
13195:
10473:(the game at each step will completely depend on the choices the players make, rather than a random factor)
10228:
5624:
459:
are different representations of the same rational number.) So strictly speaking, the surreal numbers are
12402:
Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.
11599:
the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number
10305:: given any two surreals, they are either equal, or one is greater than the other. The games have only a
9957:
The surreal exponential is essentially given by its behaviour on positive powers of ω, i.e., the function
8237:
7891:
6908:
is not an algebraic field, because it is not closed under arithmetic operations; consider ω+1, whose form
3490:
111:
14631:
14616:
14426:
13750:
12422:
Alling also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of
9812:
Conversely, dividing out the leading term of the normal form will bring any surreal number into the form
7465:. This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday
6495:
are positive infinitesimals and all the elements of the right set are positive infinities, and therefore
3136:{\displaystyle x-y=\{X_{L}\mid X_{R}\}+\{-Y_{R}\mid -Y_{L}\}=\{X_{L}-y,x-Y_{R}\mid X_{R}-y,x-Y_{L}\}\,.}
1360:
1295:
14626:
14442:
8326:
13558:
12405:
Given these axioms, Alling derives Conway's original definition of ≤ and develops surreal arithmetic.
9449:
This is well-defined for all surreal arguments (the value exists and does not depend on the choice of
7890:. The addition and multiplication of the surreal numbers associated with ordinals coincides with the
6403:
One can determine the relationship between ω and ε by multiplying particular forms of them to obtain:
2953:
which by the birthday property is a form of 1. This justifies the label used in the previous section.
14544:
14158:
14114:
13553:
10244:
10225:
3150:
Multiplication can be defined recursively as well, beginning from the special cases involving 0, the
160:
13756:
12291:
10281:
Addition, negation, and comparison are all defined the same way for both surreal numbers and games.
8951:
2410:{\displaystyle x+y=\{X_{L}\mid X_{R}\}+\{Y_{L}\mid Y_{R}\}=\{X_{L}+y,x+Y_{L}\mid X_{R}+y,x+Y_{R}\},}
14371:
14281:
14153:
11075:
10209:
9523:
1693:. These labels will also be justified by the rules for surreal addition and multiplication below.
14073:
13996:
13957:
13919:
13891:
13863:
13835:
13723:
13690:
13662:
13634:
13522:
12640:
6332:. This number is larger than zero but less than all positive dyadic fractions. It is therefore an
1578:
A second iteration of the construction rule yields the following ordering of equivalence classes:
14300:
13587:
13332:
4777:
will eventually be expressed entirely in terms of operations on numbers with birthdays less than
3151:
127:
10590:
Alternative approaches to the surreal numbers complement Conway's exposition in terms of games.
7038:{\displaystyle \omega +1=\{1,2,3,4,...\mid {}\}+\{0\mid {}\}=\{1,2,3,4,\ldots ,\omega \mid {}\}}
644:(rational numbers whose denominators are powers of 2) are contained within the surreal numbers.
14473:
13260:
12266:
12205:
12169:
12137:
12107:
10612:
8872:
8847:
8742:
8645:
8616:
8591:
6461:
This expression is only well-defined in a set theory which permits transfinite induction up to
5650:
1233:
992:
12880:. London Mathematical Society Lecture Note Series. Vol. 110. Cambridge University Press.
12561:
not have a largest element, and also the identification of a cut with the smallest element in
12045:
10545:
If a big game decomposes into two smaller games, and the small games have associated Games of
1292:
is the set of all surreal numbers that are generated by the construction rule from subsets of
14559:
14342:
14226:
14062:
13419:
13327:
13322:
12524:
12516:
12487:
8639:
8490:{\displaystyle \{x:\exists n\in \mathbb {N} :x<n\mid x:\forall n\in \mathbb {N} :x>n\}}
1510:). (The set union expression appears in our construction rule, rather than the simpler form
1245:
202:
150:. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book
13048:
Surreal
Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness
12914:
Rubinstein-Salzedo, Simon; Swaminathan, Ashvin (2015-05-19). "Analysis on
Surreal Numbers".
9995:
140:
Surreal
Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness
73:
Surreal
Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness
14401:
14396:
13979:
13712:
13358:
13312:
13209:
13203:
13118:
12727:
10616:
10485:
As soon as there are no legal moves left for a player, the game ends, and that player loses
10202:
9922:
8907:
8903:
8085:
5282:
809:
206:
152:
13242:, survey of Conway's accomplishments in the AMS Notices, with a section on surreal numbers
10518:
More generally, we can define G > H as G – H > 0, and similarly for <, = and ||.
9962:
8:
14406:
14190:
14100:
14057:
14039:
13817:
13527:
13353:
13123:
12511:
11039:
10298:
10205:
8288:
4826:
168:
6834:
6771:
14590:
14478:
14095:
13807:
13496:
13480:
13009:
12915:
12474:
10534:
10530:
10479:
Players alternate taking turns (the game may or may not allow multiple moves in a turn)
10285:
10235:, this fact characterizes the field of surcomplex numbers within any fixed set theory.
3716:
3689:
1439:
131:
104:
63:
12987:"The absolute arithmetic continuum and the unification of all numbers great and small"
12761:
12718:
8084:
To classify the "orders" of infinite and infinitesimal surreal numbers, also known as
934:
The numeric forms are placed in equivalence classes; each such equivalence class is a
14621:
14513:
14503:
14488:
14253:
14216:
14180:
14119:
14105:
13800:
13780:
13470:
13424:
13389:
13297:
13253:
13177:
13169:
13150:
13135:
13103:
13085:
13070:
13051:
13017:
12959:
12889:
12853:
12797:
12662:
12613:
12468:
11971:
The inverse map from the alternative realization to Conway's realization is given by
10284:
Every surreal number is a game, but not all games are surreal numbers, e.g. the game
10213:
9911:
8581:
5691:
as the set of all surreal numbers generated by the construction rule from subsets of
5644:
963:
are forms of the same number (lie in the same equivalence class) if and only if both
813:
460:
92:
88:
13013:
12842:(2). Warszawa: Institute of Mathematics of the Polish Academy of Sciences: 173–188.
14641:
14554:
14549:
14447:
14271:
14200:
14175:
14109:
14018:
13984:
13825:
13795:
13717:
13620:
13568:
13563:
13475:
13450:
13429:
13348:
13001:
12881:
12843:
12756:
12713:
12482:
12398:. (Alling calls a system that satisfies this axiom a "full surreal number system".)
10217:
10095:" function is not compatible with exp, since compatibility would demand a value of
9573:
8399:
7117:
with its forms in previous generations. (The ω-complete forms of real elements of
1400:
is the empty set; the only subset of the empty set is the empty set, and therefore
647:
After an infinite number of stages, infinite subsets become available, so that any
424:
100:
96:
6472:. In such a system, one can demonstrate that all the elements of the left set of
4322:, and that can be used to find more terms in a recursive fashion. For example, if
3853:
The definition of division is done in terms of the reciprocal and multiplication:
3686:, the set of numbers generated by picking all possible combinations of members of
872:
Either or both of the left and right set of a form may be the empty set. The form
14523:
14508:
14347:
14148:
14052:
13684:
13409:
13394:
13281:
13199:
12791:
12723:
12450:
11679:
11663:
11360:)) and whose range is { −, + }. "Simpler than" is very simply defined now—
10579:
10568:
10449:
10421:
8769:
8398:
of the surreal numbers with the natural ordering which is a (proper class-sized)
5955:
5810:
5278:
2872:{\displaystyle 0+y=\{{}\mid {}\}+y=\{0+Y_{L}\mid 0+Y_{R}\}=\{Y_{L}\mid Y_{R}\}=y}
2743:{\displaystyle x+0=x+\{{}\mid {}\}=\{X_{L}+0\mid X_{R}+0\}=\{X_{L}\mid X_{R}\}=x}
1241:
1060:
The recursive definition of surreal numbers is completed by defining comparison:
641:
205:
cardinals which can naturally be considered as proper classes by cutting off the
172:
13517:
13145:
A treatment of surreals based on the sign-expansion realization: Harry
Gonshor,
9108:
negative infinite the odd-numbered partial sums are strictly decreasing and the
8830:{\displaystyle \sum _{\alpha \in \mathbb {No} }r_{\alpha }\omega ^{a_{\alpha }}}
1889:
The addition, negation (additive inverse), and multiplication of surreal number
1713:
these "old" numbers, and write the ordering above using the old and new labels:
786:
collection of surreal numbers into an ordered field, so that one can talk about
14564:
14185:
14170:
13989:
13857:
13628:
13465:
13440:
13435:
13095:
10620:
9004:
8946:
5259:
1409:
consists of a single surreal form { | } lying in a single equivalence class 0.
1275:
107:
55:
9937:
an ordinal epsilon number, the set of surreal numbers with birthday less than
5928:{\displaystyle {\tfrac {1}{3}}=\{y\in S_{*}:3y<1\mid y\in S_{*}:3y>1\}.}
4528:{\displaystyle {\frac {1+(2-3)\left({\frac {1}{4}}\right)}{2}}={\frac {3}{8}}}
4450:{\displaystyle {\frac {1+(2-3)\left({\frac {1}{2}}\right)}{2}}={\frac {1}{4}}}
103:) can be realized as subfields of the surreals. The surreals also contain all
14605:
14323:
14258:
14231:
14140:
13548:
13460:
13445:
13005:
12885:
12857:
11440:
in order of simplicity (i.e., inclusion), and then write down the signs that
10470:
10306:
7781:
by the successor operation) and because it coincides with the surreal sum of
7543:
is not the successor of any ordinal. This is a surreal number with birthday
7103:
6333:
5640:
1527:
80:
51:
13228:
8729:{\textstyle \{\mathbb {No} \mid {}\}=\{\mathbb {On} \mid {}\}=\mathbb {On} }
4267:
are permitted in the formula, with any nonpositive terms being ignored (and
14498:
14483:
14221:
14023:
13543:
13455:
13399:
13161:
13043:
12934:
12684:
Alling, Norman L. (1962), "On the existence of real-closed fields that are
11079:
10221:
8361:
8240:
for ordinal numbers. This is the Conway normal form: Every surreal number
7195:
by a finite series (no longer than two, actually) of arithmetic operations
7099:
5636:
5602:
is; it is the subring of the rationals consisting of all dyadic fractions.
988:
927:
817:
707:
224:
135:
67:
39:
14452:
14358:
14047:
13829:
13404:
12423:
12101:
10302:
10232:
9558:
exp coincides with the usual exponential function on the reals (and thus
8934:
exponential, and it is this function that is meant whenever the notation
7707:
At a later stage of transfinite induction, there is a number larger than
7229:
5970:
648:
164:
142:. In his book, which takes the form of a dialogue, Knuth coined the term
43:
36:
28:
10575:
14337:
14028:
13885:
13363:
10526:
10330:
10317:
9698:{\textstyle g:\mathbb {No} _{+}\to \mathbb {No} :\alpha \mapsto \beta }
7434:
4282:, but also recursion in that the members of the left and right sets of
1560:. The smallest value of α for which a given surreal number appears in
1431:
by the ordering induced by the comparison rule on the surreal numbers.
1428:
115:
59:
13189:
12848:
12831:
11384:
as sets of ordered pairs (as a function is normally defined): Either
9910:. This stronger constraint is one of the Ressayre axioms for the real
8405:
For instance there is no least positive infinite surreal, but the gap
1661:
between −1 and 0" respectively; their equivalence classes are labeled
19:
14518:
13486:
12624:
Theorem 24.29. The surreal number system is the largest ordered field
12359:. (This axiom is often referred to as "Conway's Simplicity Theorem".)
10420:
The surreal numbers were originally motivated by studies of the game
10311:
9918:
exp satisfies all the
Ressayre axioms for the real exponential field
9576:) of exp is well defined and coincides with the inductive definition
1575:. For example, the birthday of 0 is 0, and the birthday of −1 is 1.
1229:
368:
represents a number intermediate in value between all the members of
12920:
12549:
The definition of a gap omits the conditions of a Dedekind cut that
7565:. Similarly, there are two new infinitesimal numbers in generation
6894:
is some real number that has no representation as a dyadic fraction.
5792:{\displaystyle \omega =\{S_{*}\mid {}\}=\{1,2,3,4,\ldots \mid {}\}.}
1822:
are already separated by a number created at an earlier stage, then
869:
are given as lists of elements, the braces around them are omitted.
14136:
14067:
13913:
13384:
13276:
13239:
12510:, the surreals form a proper class, rather than a set, so the term
12048:
to the reals, these axioms guarantee uniqueness up to isomorphism.
11896:
The multiplicative identity is given by the number 1 = { (0,+1) },
11412:(or both, but in this case the signs must disagree). We then have
10624:
8765:
8200:, defined on the positive surreals; it can be demonstrated that log
7236:. This can be demonstrated by exhibiting surjective mappings from
4814:
will represent the same number regardless of the choice of form of
47:
14381:
10476:
No hidden information (such as cards or tiles that a player hides)
8906:, a construction (by transfinite induction) that extends the real
8124:
then ω is "infinitely greater" than ω, in that it is greater than
5639:. With their ordering and algebraic operations they constitute an
1826:
does not represent a new number but one already constructed.) If
176:
14292:
13656:
13579:
13302:
13236:, a series of articles about surreal numbers and their variations
13187:
11908:
The map from Conway's realization to sign expansions is given by
7492:, each represented as the largest surreal number having birthday
5313:
3679:{\textstyle \left\{x'y+xy'-x'y':x'\in X_{R},~y'\in Y_{R}\right\}}
1645:
from previous generations into a left set (all numbers less than
710:). Thus the real numbers are also embedded within the surreals.
490:
are both empty, is called 0. Subsequent stages yield forms like
278:, which is common notation in other contexts, we typically write
13610:
13245:
11900:
the number 1 has domain equal to the ordinal 1, and 1(0) = +1.
10643:
10508:
If x < 0 then Right will win, regardless of who plays first.
9868:
The exponential function is much greater than any finite power
2614:{\displaystyle 0+0=\{{}\mid {}\}+\{{}\mid {}\}=\{{}\mid {}\}=0}
2531:{\displaystyle X+y=\{x'+y:x'\in X\},\quad x+Y=\{x+y':y'\in Y\}}
2167:
are drawn from generations earlier than that in which the form
13223:
Hackenstrings, and the 0.999... ?= 1 FAQ, by A. N. Walker
12096:
is a surreal number system if and only if the following hold:
10505:
If x > 0 then Left will win, regardless of who plays first.
7808:
via the construction step requires a transfinite induction on
2067:{\displaystyle -x=-\{X_{L}\mid X_{R}\}=\{-X_{R}\mid -X_{L}\},}
1033:
are the same object. This is not the case for surreal number
883:
159:
A separate route to defining the surreals began in 1907, when
12913:
12523:, or by using a form of set theory in which constructions by
10627:
is { −1, +1 }. This is equivalent to Conway's L-R sequences.
10445:
9178:
and this can be extended to the surreals with the definition
6882:
Every dyadic fraction is either greater than some element of
13110:, Chapter 4. A non-technical overview; reprint of the 1976
12089:{\textstyle \langle \mathbb {No} ,\mathrm {<} ,b\rangle }
8162:
Each power of ω also has the redeeming feature of being the
4725:{\displaystyle {\frac {1}{y}}=-\left({\frac {1}{-y}}\right)}
13130:
A detailed treatment of surreal numbers: Norman L. Alling,
12159:
11903:
10501:. We can classify all Games into four classes as follows:
10415:
7507:. There is another positive infinite number in generation
6140:
among the reals. Consider the smallest positive number in
4561:
3931:
13191:
Homotopy Type Theory: Univalent Foundations of Mathematics
8159:
so they behave the way one would expect powers to behave.
5330:
The set of all surreal numbers that are generated in some
2078:
of numbers is given by the set of the negated elements of
10630:
Define the binary predicate "simpler than" on numbers by
10412:. Note that "=" here means equality, not identity.
9075:{\displaystyle \exp x=\sum _{n\geq 0}{\frac {x^{n}}{n!}}}
9003:, and where defined it necessarily agrees with any other
8287:
Looked at in this manner, the surreal numbers resemble a
8079:
1785:
represents a number inherited from an earlier generation
10301:, but the class of games does not. The surreals have a
6518:
is the oldest positive finite number, 1. Consequently,
2242:
The definition of addition is also a recursive formula:
1613:< { 1 | } = { 0, 1 | } = { −1, 1 | } = { −1, 0, 1 | }
379:
Different subsets may end up defining the same number:
247:
in many other mathematical contexts, is instead written
7416:; and so forth. This maps a nonempty open interval of
7098:
This construction of the real numbers differs from the
4802:(equivalence classes of forms): the result of negating
12294:
12269:
12208:
12172:
12140:
12110:
12057:
11595:
The additive identity is given by the number 0 = { },
9650:
9526:
8954:
8875:
8850:
8745:
8673:
8648:
8619:
8594:
8539:
8505:
8370:
8329:
8204:
maps the positive surreals onto the surreals and that
6837:
6774:
6257:
6242:
6227:
6070:
6055:
6040:
6013:
5998:
5837:
5653:
5351:
3719:
3692:
3570:
3493:
1862:. In other words, it lies in the equivalence class in
1363:
1298:
14076:
13999:
13960:
13922:
13894:
13866:
13838:
13759:
13726:
13693:
13665:
13637:
13229:
A gentle yet thorough introduction by Claus Tøndering
10254:. All games are constructed according to this rule:
9998:
9965:
9572:
infinitesimal, the value of the formal power series (
9186:
9021:
8778:
8413:
7814:
6914:
6157:
5979:
5835:
5717:
5407:
4682:
4541:
4463:
4385:
3911:
3861:
3160:
2967:
2755:
2626:
2547:
2425:
2248:
2177:
2088:
1981:
1830:
represents a number from any generation earlier than
1583:{ | −1 } = { | −1, 0 } = { | −1, 1 } = { | −1, 0, 1 }
12825:
12464:
10553:, then the big game will have an associated Game of
9644:. In fact there is an inductively defined bijection
7850:{\displaystyle \bigcup _{k<\omega }S_{\omega +k}}
7804:. It is the second limit ordinal; reaching it from
5813:. For example, the ω-complete form of the fraction
5809:
also contains objects that can be identified as the
5584:
is closed under addition and multiplication (except
3894:{\displaystyle {\frac {x}{y}}=x\cdot {\frac {1}{y}}}
8573:{\textstyle \mathbb {On} =\{\mathbb {No} \mid {}\}}
7477:Continuing to perform transfinite induction beyond
2961:Subtraction is defined with addition and negation:
1244:; a wider universe is reachable given some form of
876:with both left and right set empty is also written
539:section below). Similarly, representations such as
14084:
14007:
13968:
13930:
13902:
13874:
13846:
13769:
13734:
13701:
13673:
13645:
13208:The surreal numbers are studied in the context of
13132:Foundations of Analysis over Surreal Number Fields
12956:Foundations of Analysis over Surreal Number Fields
12311:
12280:
12219:
12183:
12151:
12121:
12088:
10514:If x || 0 then the player who goes first will win.
10511:If x = 0 then the player who goes second will win.
10019:
9980:
9697:
9544:
9439:
9074:
8971:
8918:) to the surreals was carried through by Gonshor.
8886:
8861:
8829:
8756:
8728:
8659:
8630:
8613:denotes the class of ordinal numbers, and because
8605:
8572:
8525:
8489:
8390:
8352:
8170:there will always exist some positive real number
8088:classes, Conway associated to each surreal number
7875:, but the surreal sum is commutative and produces
7849:
7037:
6849:
6786:
6324:
6097:
5927:
5791:
5664:
5554:
5393:
4724:
4642:
4527:
4449:
4242:
3893:
3732:
3705:
3678:
3548:
3479:
3135:
2871:
2742:
2613:
2530:
2409:
2228:{\displaystyle -0=-\{{}\mid {}\}=\{{}\mid {}\}=0.}
2227:
2137:This formula involves the negation of the surreal
2127:
2066:
1392:
1327:
12638:
1754:establish that these are forms of 3 by using the
833:is a pair of sets of surreal numbers, called its
14603:
13147:An Introduction to the Theory of Surreal Numbers
12878:An Introduction to the Theory of Surreal Numbers
11376:. The total ordering is defined by considering
9479:exp is a strictly increasing positive function,
8580:is larger than all surreal numbers. (This is an
4302:itself. 0 is always a member of the left set of
702:is the set of all dyadic rationals greater than
10567:A game composed of smaller games is called the
10482:Every game must end in a finite number of moves
10158:, giving an interpretation to expressions like
7554:on the basis that it coincides with the sum of
3556:that appears in the left set of the product of
1522:, so that the definition also makes sense when
58:than any positive real number. Research on the
13119:"Infinity Plus One, and Other Surreal Numbers"
12981:
12832:"Fields of surreal numbers and exponentiation"
12634:
12632:
12519:, yielding a set with the cardinality of some
11447:
7270:is routine; map numbers less than or equal to
2171:first occurs, and observing the special case:
1208:. That is, every element in the right part of
14308:
13595:
13261:
9705:whose inverse can also be defined inductively
7777:(the first ordinal number not reachable from
4535:will be a right term. Continuing, this gives
3740:, and substituting them into the expression.
1758:, which is a consequence of the rules above.
1162:. That is, every element in the left part of
687:is the set of all dyadic rationals less than
12083:
12058:
9431:
9199:
8712:
8696:
8690:
8674:
8567:
8551:
8484:
8414:
7032:
6991:
6985:
6974:
6968:
6927:
6794:is greater than or equal to all elements of
6319:
6282:
6203:
6164:
5919:
5851:
5783:
5748:
5742:
5724:
5545:
5514:
5490:
5459:
5435:
5429:
3233:
3207:
3204:
3178:
3126:
3050:
3044:
3012:
3006:
2980:
2860:
2834:
2828:
2790:
2778:
2768:
2731:
2705:
2699:
2661:
2655:
2645:
2602:
2592:
2586:
2576:
2570:
2560:
2525:
2491:
2472:
2438:
2401:
2325:
2319:
2293:
2287:
2261:
2216:
2206:
2200:
2190:
2119:
2098:
2058:
2026:
2020:
1994:
1873:that is a superset of the representation of
1649:) and a right set (all numbers greater than
1598:< { | } = { −1 | } = { | 1 } = { −1 | 1 }
12629:
10837:if and only if one of the following holds:
10611:of a surreal number, a surreal number is a
10585:
10143:A general exponentiation can be defined as
9475:Using this definition, the following hold:
8364:, but we can still talk about a completion
8132:. Powers of ω also satisfy the conditions
1700:of induction may be characterized by their
884:Numeric forms and their equivalence classes
23:A visualization of the surreal number tree.
14315:
14301:
14277:
13602:
13588:
13268:
13254:
13188:The Univalent Foundations Program (2013).
12958:. Mathematics Studies 141. North-Holland.
12821:
12819:
12817:
12815:
12813:
8764:can be equipped with a topology where the
8526:{\textstyle \mathbb {No} _{\mathfrak {D}}}
8391:{\textstyle \mathbb {No} _{\mathfrak {D}}}
8116:range over the positive real numbers. If
8034:. It can be identified as the product of
8006:where on the right hand side the notation
6861:and less than or equal to all elements of
6129:; but there are other non-real numbers in
5643:, with the caveat that they do not form a
1037:, but is true by construction for surreal
352:are strictly less than all the members of
299:
14078:
14001:
13962:
13924:
13896:
13868:
13840:
13728:
13695:
13667:
13639:
12949:
12947:
12945:
12943:
12919:
12847:
12760:
12733:
12717:
12444:
12305:
12302:
12274:
12271:
12213:
12210:
12177:
12174:
12145:
12142:
12115:
12112:
12065:
12062:
10455:We consider games with these properties:
9679:
9676:
9662:
9659:
9532:
9529:
8965:
8962:
8880:
8877:
8855:
8852:
8794:
8791:
8750:
8747:
8722:
8719:
8703:
8700:
8681:
8678:
8653:
8650:
8624:
8621:
8599:
8596:
8558:
8555:
8544:
8541:
8511:
8508:
8468:
8433:
8376:
8373:
8346:
8343:
7793:because it coincides with the product of
7281:) to 0, numbers greater than or equal to
7129:construction; they are merely the subset
5658:
5655:
5394:{\textstyle S_{*}=\bigcup _{n\in N}S_{n}}
4605:
4599:
4093:
4092:
4086:
4085:
3129:
1942:are defined by three recursive formulas.
1499:(as supersets of their representation in
1267:, in which 0 consists of the single form
121:
110:; the arithmetic on them is given by the
13225:, an archive of the disappeared original
13082:Winning Ways for Your Mathematical Plays
12977:
12975:
12935:Surreal vectors and the game of Cutblock
12909:
12907:
12905:
12871:
12869:
12867:
12601:
12599:
11904:Correspondence with Conway's realization
10416:Application to combinatorial game theory
10216:of the field generated by extending the
8844:decreasing and having no lower bound in
7472:
6594:, exactly one of the following is true:
3743:For example, to show that the square of
2141:appearing in the left and right sets of
348:of numbers such that all the members of
18:
12875:
12810:
12785:
12783:
12781:
12779:
12777:
12527:stop at some countable ordinal such as
12408:
8897:
7203:is strictly smaller than the subset of
6749:equals the oldest such dyadic fraction
1795:if and only if there is some number in
995:, i.e., it must have the property that
427:are defined as quotients of integers:
175:introduced certain ordered sets called
14604:
12953:
12940:
12789:
12739:
12683:
12605:
11082:(except that < is a proper class).
9594:is given by power series expansion of
8298:
8080:Powers of ω and the Conway normal form
1696:The equivalence classes at each stage
1541:that are a superset of some number in
223:In the context of surreal numbers, an
14427:Infinitesimal strain theory (physics)
14296:
13583:
13249:
12972:
12902:
12864:
12609:An Invitation to Abstract Mathematics
12596:
12166:is called the "birthday function" on
12038:
10165:
9640:is a strictly increasing function of
8921:
7303:(mapping the infinitesimal neighbors
6375:; adding them to any dyadic fraction
5253:
3549:{\textstyle X_{R}y+xY_{R}-X_{R}Y_{R}}
1806:that is greater than all elements of
930:≤ given by the comparison rule below.
918:is the empty set and each element of
423:. (A similar phenomenon occurs when
13318:Hilbert's paradox of the Grand Hotel
13234:Good Math, Bad Math: Surreal Numbers
12774:
12508:von Neumann–Bernays–Gödel set theory
12455:von Neumann–Bernays–Gödel set theory
9921:The surreals with exponential is an
9712:is "pure infinite" with normal form
1761:
1631:generations into two non-empty sets.
713:There are also representations like
207:cumulative hierarchy of the universe
118:to the maximal class surreal field.
85:von Neumann–Bernays–Gödel set theory
54:, respectively larger or smaller in
13791:Set-theoretically definable numbers
13058:. More information can be found at
12742:"Conway's Field of surreal numbers"
10208:(except for being a proper class),
8517:
8382:
1834:, there is a least such generation
1393:{\textstyle \bigcup _{i<0}S_{i}}
1328:{\textstyle \bigcup _{i<n}S_{i}}
1052:is a form of the surreal number 0.
403:may define the same number even if
13:
14322:
13762:
13609:
13100:Penrose Tiles to Trapdoor Ciphers,
13037:
12506:In the original formulation using
10396:, then it is not always true that
10201:. The surcomplex numbers form an
9552:) and has a well-defined inverse,
9092:and include all partial sums; for
9010:
8739:With a bit of set-theoretic care,
8458:
8423:
8353:{\textstyle L\cup R=\mathbb {No} }
7933:that is infinite but smaller than
7292:) to 1, and numbers between ε and
7197:including multiplicative inversion
7081:, and the infinitesimal neighbors
6542:. Some authors systematically use
5401:. One may form the three classes
4798:The operations can be extended to
1846:as its birthday that lies between
1810:and less than all elements of the
146:for what Conway had called simply
16:Generalization of the real numbers
14:
14653:
14529:Transcendental law of homogeneity
14422:Constructive nonstandard analysis
14366:The Method of Mechanical Theorems
14353:Criticism of nonstandard analysis
13513:Differential geometry of surfaces
13240:Conway's Mathematics after Conway
13216:
13168:, New York: Polity Press, 2008,
12762:10.1090/s0002-9947-1985-0766225-7
12719:10.1090/S0002-9947-1962-0146089-X
12660:
12639:O'Connor, J.J.; Robertson, E.F.,
11705:, is defined by induction on dom(
11468:, is defined by induction on dom(
11332:as particular cases of surreals.
10593:
10321:(incomparable with zero, such as
10138:
9119:The relations that hold for real
8276:is a nonzero real number and the
7429:, monotonically. The residue of
6647:is greater than every element of
6633:equals the smallest such integer
6625:is greater than every element of
6555:
3145:
2128:{\displaystyle -S=\{-s:s\in S\}.}
1251:
1044:The equivalence class containing
922:is greater than every element of
332:. (Both may hold, in which case
14380:
14276:
13308:Controversy over Cantor's theory
13275:
12467:
11117:), there exists a unique number
10533:. An example of a fuzzy game is
9590:infinitesimally close to 1, log
8927:
8292:
7340:, map the (open) central third (
7091:of each nonzero dyadic fraction
6857:is greater than all elements of
6683:equals the largest such integer
5954:Not only do all the rest of the
5617:
3811:= { 0 | 1 } ⋅ { 0 | 1 } = { 0 |
3154:1, and its additive inverse −1:
1412:For every finite ordinal number
475:that designate the same number.
260:comma. For example, instead of
14412:Synthetic differential geometry
13369:Synthetic differential geometry
13102:W. H. Freeman & Co., 1989,
12937:, James Propp, August 22, 1994.
12928:
12790:Conway, John H. (2000-12-11) .
12577:
12568:
12543:
12312:{\textstyle z\in \mathbb {No} }
12255:(using Alling's terminology, 〈
11232:is the simplest number between
10685:if one of the following holds:
8972:{\textstyle x\in \mathbb {No} }
7892:natural sum and natural product
7539:is not an ordinal; the ordinal
4329:}, then we know a left term of
2478:
1593:< { −1 | 0 } = { −1 | 0, 1 }
803:
463:of representations of the form
13770:{\displaystyle {\mathcal {P}}}
12994:The Bulletin of Symbolic Logic
12677:
12654:
12534:
12521:strongly inaccessible cardinal
12500:
12417:
12288:). Then there exists a unique
11392:, or else the surreal number
11078:). This means that < is a
10598:
10333:for the first player to move.
10091:This shows that the "power of
10008:
10002:
9975:
9969:
9689:
9672:
9545:{\textstyle \mathbb {No} _{+}}
9413:
9393:
9360:
9340:
9298:
9278:
9247:
9227:
8319:is less than every element of
7488:produces more ordinal numbers
7045:does not lie in any number in
6798:and less than all elements of
6741:and less than all elements of
6737:(greater than all elements of
6697:is less than every element of
6675:is less than every element of
6605:are both empty, in which case
5056:= { −4 < −3 < ... < −
4764:
4485:
4473:
4407:
4395:
4345:will be 0. This in turn means
4194:
4175:
4122:
4103:
3968:
3949:
2956:
1603:< { 0 | 1 } = { −1, 0 | 1 }
1048:is labeled 0; in other words,
482:. This representation, where
213:
1:
14581:Analyse des Infiniment Petits
14417:Smooth infinitesimal analysis
14125:Plane-based geometric algebra
12590:
10269:are two sets of games then {
9925:of the real exponential field
8902:Based on unpublished work by
8196:is the "base ω logarithm" of
8188:is "infinitely smaller" than
7767:both because its birthday is
7522:− 1 = { 0, 1, 2, 3, 4, ... |
6886:or less than some element of
5573:is the union. No individual
5281:, i.e., can be written as an
1884:
1228:This group of definitions is
14085:{\displaystyle \mathbb {S} }
14008:{\displaystyle \mathbb {C} }
13969:{\displaystyle \mathbb {R} }
13931:{\displaystyle \mathbb {O} }
13903:{\displaystyle \mathbb {H} }
13875:{\displaystyle \mathbb {C} }
13847:{\displaystyle \mathbb {R} }
13735:{\displaystyle \mathbb {A} }
13702:{\displaystyle \mathbb {Q} }
13674:{\displaystyle \mathbb {Z} }
13646:{\displaystyle \mathbb {N} }
13415:Cardinality of the continuum
13196:Institute for Advanced Study
13060:the book's official homepage
10444:for recreational games like
10428:for the mathematical object
10037:for some number of levels).
7915:, there is a surreal number
7247:to the closed unit interval
5935:The product of this form of
4833:and multiplicative identity
4379:is a right term. This means
2915:= { 0 | 1 } + { 0 | 1 } = {
2074:where the negation of a set
1455:, since every valid form in
1232:, and requires some form of
1223:
7:
12740:Alling, Norman (Jan 1985),
12460:
12362:Furthermore, if an ordinal
12162:the class of all ordinals (
11448:Addition and multiplication
11228:by transfinite induction.
10781:)} ∪ { α : α < dom(
10531:positive, negative, or zero
9952:
9086:positive these are denoted
8315:such that every element of
8244:may be uniquely written as
7757:This number may be labeled
7214:identified with the reals.
6831:, but some dyadic fraction
6768:, but some dyadic fraction
5675:
5266:, all numbers generated in
4457:is a left term. This means
3848:
2237:
1950:Negation of a given number
1945:
218:
10:
14658:
13378:Formalizations of infinity
12954:Alling, Norman L. (1987).
12281:{\textstyle \mathbb {No} }
12220:{\textstyle \mathbb {No} }
12184:{\textstyle \mathbb {No} }
12152:{\textstyle \mathbb {No} }
12122:{\textstyle \mathbb {No} }
11682:if and only if 0 < dom(
11666:if and only if 0 < dom(
10603:In what is now called the
10242:
9871:For any positive infinite
9774:, the inverse is given by
9520:exp is a surjection (onto
9470:
8945:is a dyadic fraction, the
8938:is used in the following.
8887:{\textstyle \mathbb {On} }
8862:{\textstyle \mathbb {No} }
8757:{\textstyle \mathbb {No} }
8660:{\textstyle \mathbb {No} }
8631:{\textstyle \mathbb {On} }
8606:{\textstyle \mathbb {On} }
8038:and the form { 0 | 1 } of
7789:; it may also be labeled 2
7559:= { 0, 1, 2, 3, 4, ... | }
6668:is empty and some integer
6618:is empty and some integer
6371:and its additive inverse −
5665:{\textstyle \mathbb {No} }
1608:< { 0 | } = { −1, 0 | }
14612:Combinatorial game theory
14573:
14545:Gottfried Wilhelm Leibniz
14537:
14466:
14435:
14389:
14378:
14330:
14267:
14209:
14135:
14115:Algebra of physical space
14037:
13945:
13816:
13618:
13554:Gottfried Wilhelm Leibniz
13536:
13505:
13377:
13341:
13290:
13084:, vol. 1, 2nd ed., 2001,
12796:(2 ed.). CRC Press.
12327:) is minimal and for all
11658:It follows that a number
11240:. Let the unique number
10245:Combinatorial game theory
10226:algebraically independent
7255:and vice versa. Mapping
7178:. By demonstrating that
7056:. The maximal subset of
6722:are both non-empty, and:
4829:, with additive identity
4806:or adding or multiplying
1838:, and exactly one number
1588:< { | 0 } = { | 0, 1 }
1166:is strictly smaller than
1025:are both true) only when
718:{ 0, 1, 2, 3, ... | } =
533:{ | −2 } = −3
526:{ | −1 } = −2
14171:Extended complex numbers
14154:Extended natural numbers
13046:'s original exposition:
12886:10.1017/CBO9780511629143
12493:
10586:Alternative realizations
10238:
10193:are surreal numbers and
10174:is a number of the form
8174:and some surreal number
7798:= { 1, 2, 3, 4, ... | }
7717:for all natural numbers
7412:) of the upper third to
7313:of each dyadic fraction
7144:containing all elements
6693:is empty and no integer
6643:is empty and no integer
3564:. This is understood as
1781:occurring in generation
1477:, all of the numbers in
1466:is also a valid form in
1212:is strictly larger than
1055:
824:
42:containing not only the
13559:August Ferdinand Möbius
13342:Branches of mathematics
13333:Paradoxes of set theory
12876:Gonshor, Harry (1986).
12836:Fundamenta Mathematicae
12749:Trans. Amer. Math. Soc.
12706:Trans. Amer. Math. Soc.
12263:〉 is a "Conway cut" of
10670:(α) for all α < dom(
10231:elements. Up to field
8128:ω for all real numbers
7943:for any natural number
7911:for any natural number
7299:to their equivalent in
6548:in place of the symbol
6107:The only infinities in
5969:; the remaining finite
3152:multiplicative identity
1814:. (In other words, if
1041:(equivalence classes).
910:if the intersection of
841:. A form with left set
810:constructed inductively
372:and all the members of
300:Outline of construction
167:as a generalization of
14474:Standard part function
14227:Transcendental numbers
14086:
14063:Hyperbolic quaternions
14009:
13970:
13932:
13904:
13876:
13848:
13771:
13736:
13703:
13675:
13647:
13006:10.2178/bsl/1327328438
12557:be non-empty and that
12445:Relation to hyperreals
12313:
12282:
12221:
12185:
12153:
12123:
12090:
11259:, define its left set
11220:is constructible from
11042:, and for all numbers
10821:). Then, for numbers
10725:there exists a number
10197:is the square root of
10160:2 = exp(ω · log 2) = ω
10021:
10020:{\displaystyle g(a)=a}
9982:
9699:
9605:For positive infinite
9546:
9441:
9076:
8973:
8888:
8863:
8831:
8758:
8730:
8661:
8632:
8607:
8574:
8527:
8491:
8392:
8354:
7851:
7039:
6851:
6823:lies strictly between
6788:
6760:lies strictly between
6729:is "strictly between"
6336:number, often labeled
6326:
6099:
5973:do too. For example,
5929:
5793:
5666:
5556:
5395:
4863:= { −1 < 0 < 1 }
4726:
4644:
4529:
4451:
4244:
3895:
3734:
3707:
3680:
3550:
3481:
3137:
2873:
2744:
2615:
2532:
2411:
2229:
2129:
2068:
1552:are said to have been
1394:
1329:
1257:There is a generation
1234:mathematical induction
654:can be represented by
235:, which is written as
122:History of the concept
24:
14560:Augustin-Louis Cauchy
14372:Cavalieri's principle
14159:Extended real numbers
14087:
14010:
13980:Split-complex numbers
13971:
13933:
13905:
13877:
13849:
13772:
13737:
13713:Constructible numbers
13704:
13676:
13648:
13523:Möbius transformation
13420:Dedekind-infinite set
13328:Paradoxes of infinity
13323:Infinity (philosophy)
12606:Bajnok, Béla (2013).
12525:transfinite recursion
12517:Grothendieck universe
12488:Non-standard analysis
12314:
12283:
12222:
12186:
12154:
12124:
12091:
11085:For sets of numbers,
11038:The relation < is
10022:
9983:
9700:
9547:
9442:
9096:negative but finite,
9077:
8974:
8889:
8864:
8832:
8759:
8731:
8662:
8633:
8608:
8575:
8528:
8492:
8393:
8355:
8072:is the limit ordinal
7852:
7547:+1, which is labeled
7473:Transfinite induction
7420:onto each element of
7380:; the central third (
7074:, the infinitesimals
7040:
6852:
6789:
6725:Some dyadic fraction
6379:produces the numbers
6352:) is the same as the
6327:
6100:
5930:
5794:
5667:
5557:
5396:
4917:= { −3 < −2 < −
4874:= { −2 < −1 < −
4727:
4645:
4530:
4452:
4245:
3896:
3735:
3708:
3681:
3551:
3482:
3138:
2874:
2745:
2616:
2533:
2412:
2230:
2130:
2069:
1634:Every surreal number
1395:
1330:
1246:transfinite induction
1123:if and only if both:
989:ordering relationship
203:strongly inaccessible
52:infinitesimal numbers
22:
14637:Nonstandard analysis
14402:Nonstandard calculus
14397:Nonstandard analysis
14191:Supernatural numbers
14101:Multicomplex numbers
14074:
14058:Dual-complex numbers
13997:
13958:
13920:
13892:
13864:
13836:
13818:Composition algebras
13786:Arithmetical numbers
13757:
13724:
13691:
13663:
13635:
13359:Nonstandard analysis
13210:homotopy type theory
13067:On Numbers And Games
12826:van den Dries, Lou;
12793:On Numbers and Games
12695:-sets of power
12409:Simplicity hierarchy
12292:
12267:
12206:
12170:
12138:
12108:
12055:
11404:is in the domain of
10765:Equivalently, let δ(
10440:, and the lowercase
10203:algebraically closed
9996:
9981:{\displaystyle g(a)}
9963:
9923:elementary extension
9648:
9613:is infinite as well
9524:
9184:
9019:
8952:
8908:exponential function
8898:Exponential function
8873:
8848:
8776:
8743:
8671:
8646:
8617:
8592:
8537:
8533:. Similarly the gap
8503:
8411:
8368:
8327:
7812:
7232:as the real numbers
7199:, one can show that
6912:
6835:
6772:
6389:, which also lie in
6155:
5977:
5833:
5715:
5651:
5405:
5349:
5283:irreducible fraction
4680:
4539:
4461:
4383:
3909:
3859:
3717:
3690:
3568:
3491:
3158:
2965:
2753:
2624:
2545:
2423:
2246:
2175:
2086:
1979:
1361:
1296:
1063:Given numeric forms
808:Surreal numbers are
153:On Numbers and Games
14586:Elementary Calculus
14467:Individual concepts
14407:Internal set theory
14096:Split-biquaternions
13808:Eisenstein integers
13746:Closed-form numbers
13528:Riemannian manifold
13497:Transfinite numbers
13354:Internal set theory
13112:Scientific American
12134:is a function from
10459:Two players (named
9560:exp 0 = 1, exp 1 =
8299:Gaps and continuity
8092:the surreal number
8054:. The birthday of
7532:The surreal number
6850:{\textstyle y\in R}
6819:No dyadic fraction
6787:{\textstyle y\in L}
6756:No dyadic fraction
944:equivalence classes
926:, according to the
814:equivalence classes
640:arise, so that the
461:equivalence classes
169:formal power series
83:. If formulated in
14632:John Horton Conway
14617:Mathematical logic
14479:Transfer principle
14343:Leibniz's notation
14254:Profinite integers
14217:Irrational numbers
14082:
14005:
13966:
13928:
13900:
13872:
13844:
13801:Gaussian rationals
13781:Computable numbers
13767:
13732:
13699:
13671:
13643:
13481:Sphere at infinity
13432:(Complex infinity)
13166:Number and Numbers
12475:Mathematics portal
12309:
12278:
12227:such that for all
12217:
12181:
12149:
12119:
12086:
12046:axiomatic approach
12039:Axiomatic approach
10348:are surreals, and
10166:Surcomplex numbers
10017:
9978:
9695:
9542:
9437:
9072:
9049:
8969:
8922:Other exponentials
8884:
8859:
8827:
8799:
8754:
8726:
8657:
8628:
8603:
8570:
8523:
8487:
8388:
8350:
8289:power series field
8238:Cantor normal form
7847:
7830:
7186:is reachable from
7169:, both drawn from
7035:
6847:
6784:
6344:-complete form of
6322:
6266:
6251:
6236:
6095:
6079:
6064:
6049:
6022:
6007:
5925:
5846:
5789:
5662:
5552:
5550:
5391:
5380:
5345:may be denoted as
5254:Arithmetic closure
4722:
4640:
4525:
4447:
4240:
3891:
3733:{\textstyle Y_{R}}
3730:
3706:{\textstyle X_{R}}
3703:
3676:
3546:
3477:
3475:
3133:
2869:
2740:
2611:
2528:
2407:
2225:
2125:
2064:
1858:is a form of this
1390:
1379:
1325:
1314:
955:Two numeric forms
706:(reminiscent of a
519:{ | 0 } = −1
132:John Horton Conway
112:natural operations
89:rational functions
64:John Horton Conway
25:
14627:Real closed field
14599:
14598:
14514:Law of continuity
14504:Levi-Civita field
14489:Increment theorem
14448:Hyperreal numbers
14290:
14289:
14201:Superreal numbers
14181:Levi-Civita field
14176:Hyperreal numbers
14120:Spacetime algebra
14106:Geometric algebra
14019:Bicomplex numbers
13985:Split-quaternions
13826:Division algebras
13796:Gaussian integers
13718:Algebraic numbers
13621:definable numbers
13577:
13576:
13471:Point at infinity
13451:Hyperreal numbers
13425:Directed infinity
13390:Absolute infinite
13313:Galileo's paradox
13298:Ananta (infinite)
13194:. Princeton, NJ:
13069:, 2nd ed., 2001,
12849:10.4064/fm167-2-3
12663:"Surreal Numbers"
11408:or the domain of
11050:, exactly one of
10805:if and only if δ(
10258:Construction rule
10214:algebraic closure
10172:surcomplex number
9912:exponential field
9070:
9034:
8779:
8588:in the surreals;
7815:
7165:and some nonzero
7104:standard analysis
6745:), in which case
6265:
6250:
6235:
6078:
6063:
6048:
6021:
6006:
5845:
5365:
5262:(finite ordinal)
4716:
4691:
4627:
4614:
4591:
4578:
4550:
4523:
4510:
4500:
4445:
4432:
4422:
4233:
4210:
4161:
4138:
4083:
4060:
4007:
3984:
3920:
3889:
3870:
3649:
1762:Birthday property
1756:birthday property
1717:−2 < −1 < −
1364:
1357:, the expression
1299:
1281:, the generation
889:Construction rule
101:hyperreal numbers
97:superreal numbers
93:Levi-Civita field
14649:
14555:Pierre de Fermat
14550:Abraham Robinson
14390:Related branches
14384:
14317:
14310:
14303:
14294:
14293:
14280:
14279:
14247:
14237:
14149:Cardinal numbers
14110:Clifford algebra
14091:
14089:
14088:
14083:
14081:
14053:Dual quaternions
14014:
14012:
14011:
14006:
14004:
13975:
13973:
13972:
13967:
13965:
13937:
13935:
13934:
13929:
13927:
13909:
13907:
13906:
13901:
13899:
13881:
13879:
13878:
13873:
13871:
13853:
13851:
13850:
13845:
13843:
13776:
13774:
13773:
13768:
13766:
13765:
13741:
13739:
13738:
13733:
13731:
13708:
13706:
13705:
13700:
13698:
13685:Rational numbers
13680:
13678:
13677:
13672:
13670:
13652:
13650:
13649:
13644:
13642:
13604:
13597:
13590:
13581:
13580:
13569:Abraham Robinson
13564:Bernhard Riemann
13483:(Kleinian group)
13476:Regular cardinal
13430:Division by zero
13410:Cardinal numbers
13349:Complex analysis
13284:
13270:
13263:
13256:
13247:
13246:
13212:in section 11.6.
13207:
13127:, December 1995.
13032:
13031:
13029:
13028:
13022:
13016:. Archived from
12991:
12979:
12970:
12969:
12951:
12938:
12932:
12926:
12925:
12923:
12911:
12900:
12899:
12873:
12862:
12861:
12851:
12830:(January 2001).
12823:
12808:
12807:
12787:
12772:
12771:
12770:
12769:
12764:
12746:
12737:
12731:
12730:
12721:
12703:
12694:
12681:
12675:
12674:
12672:
12670:
12658:
12652:
12651:
12650:
12649:
12642:Conway Biography
12636:
12627:
12626:
12603:
12584:
12581:
12575:
12572:
12566:
12547:
12541:
12538:
12532:
12504:
12483:Hyperreal number
12477:
12472:
12471:
12440:
12366:is greater than
12346:
12336:
12318:
12316:
12315:
12310:
12308:
12287:
12285:
12284:
12279:
12277:
12246:
12236:
12226:
12224:
12223:
12218:
12216:
12190:
12188:
12187:
12182:
12180:
12158:
12156:
12155:
12150:
12148:
12128:
12126:
12125:
12120:
12118:
12095:
12093:
12092:
12087:
12076:
12068:
11891:
11810:
11697:of two numbers,
11607:, given by dom(−
11590:
11541:
11460:of two numbers,
11364:is simpler than
11267:) and right set
11244:be denoted by σ(
11208:is simpler than
11074:, holds (law of
10797:(α) }), so that
10741:is simpler than
10733:is simpler than
10710:is simpler than
10692:is simpler than
10634:is simpler than
10562:
10529:, as opposed to
10439:
10411:
10395:
10373:
10357:
10324:
10218:rational numbers
10200:
10196:
10184:
10161:
10157:
10036:
10028:
10026:
10024:
10023:
10018:
9989:
9987:
9985:
9984:
9979:
9890:For any integer
9856:
9844:
9801:
9773:
9761:
9750:
9738:
9704:
9702:
9701:
9696:
9682:
9671:
9670:
9665:
9600:
9574:Taylor expansion
9564:
9555:
9551:
9549:
9548:
9543:
9541:
9540:
9535:
9517:
9496:
9446:
9444:
9443:
9438:
9430:
9429:
9405:
9404:
9392:
9387:
9386:
9368:
9367:
9352:
9351:
9339:
9334:
9333:
9315:
9314:
9296:
9295:
9274:
9273:
9255:
9254:
9245:
9244:
9223:
9222:
9174:
9148:
9128:
9081:
9079:
9078:
9073:
9071:
9069:
9061:
9060:
9051:
9048:
9007:that can exist.
9002:
8988:
8978:
8976:
8975:
8970:
8968:
8893:
8891:
8890:
8885:
8883:
8868:
8866:
8865:
8860:
8858:
8836:
8834:
8833:
8828:
8826:
8825:
8824:
8823:
8809:
8808:
8798:
8797:
8770:Cauchy sequences
8763:
8761:
8760:
8755:
8753:
8735:
8733:
8732:
8727:
8725:
8711:
8706:
8689:
8684:
8666:
8664:
8663:
8658:
8656:
8637:
8635:
8634:
8629:
8627:
8612:
8610:
8609:
8604:
8602:
8587:
8579:
8577:
8576:
8571:
8566:
8561:
8547:
8532:
8530:
8529:
8524:
8522:
8521:
8520:
8514:
8496:
8494:
8493:
8488:
8471:
8436:
8400:linear continuum
8397:
8395:
8394:
8389:
8387:
8386:
8385:
8379:
8359:
8357:
8356:
8351:
8349:
8314:
8192:. The exponent
8187:
8154:
8152:
8151:
8148:
8145:
8071:
8069:
8068:
8065:
8062:
8053:
8051:
8050:
8047:
8044:
8033:
8014:is used to mean
8002:
7986:
7984:
7983:
7980:
7977:
7964:
7962:
7961:
7958:
7955:
7942:
7932:
7930:
7929:
7926:
7923:
7910:
7889:
7870:
7856:
7854:
7853:
7848:
7846:
7845:
7829:
7803:
7799:
7776:
7766:
7716:
7702:
7696:
7694:
7693:
7690:
7687:
7676:
7674:
7673:
7670:
7667:
7651:
7649:
7648:
7645:
7642:
7631:
7629:
7628:
7625:
7622:
7611:
7609:
7608:
7605:
7602:
7571:
7564:
7560:
7553:
7538:
7527:
7513:
7506:
7495:
7491:
7487:
7468:
7464:
7453:
7441:
7433:consists of the
7432:
7428:
7419:
7415:
7411:
7409:
7408:
7405:
7402:
7395:
7393:
7392:
7389:
7386:
7379:
7375:
7371:
7369:
7368:
7365:
7362:
7355:
7353:
7352:
7349:
7346:
7339:
7328:
7324:
7320:
7316:
7312:
7302:
7298:
7291:
7287:
7280:
7273:
7269:
7265:
7254:
7250:
7246:
7235:
7227:
7213:
7202:
7194:
7185:
7181:
7177:
7168:
7164:
7160:
7147:
7143:
7132:
7127:
7116:
7094:
7090:
7080:
7073:
7066:
7055:
7044:
7042:
7041:
7036:
7031:
6984:
6967:
6907:
6893:
6890:, in which case
6889:
6885:
6878:
6868:
6865:, in which case
6864:
6860:
6856:
6854:
6853:
6848:
6830:
6826:
6822:
6815:
6805:
6802:, in which case
6801:
6797:
6793:
6791:
6790:
6785:
6767:
6763:
6759:
6752:
6748:
6744:
6740:
6736:
6732:
6728:
6721:
6717:
6711:
6704:
6701:, in which case
6700:
6696:
6692:
6686:
6682:
6679:, in which case
6678:
6674:
6667:
6661:
6654:
6651:, in which case
6650:
6646:
6642:
6636:
6632:
6629:, in which case
6628:
6624:
6617:
6611:
6604:
6600:
6593:
6582:
6551:
6547:
6541:
6536:
6534:
6533:
6528:
6525:
6517:
6494:
6471:
6456:
6399:
6388:
6378:
6374:
6370:
6366:
6355:
6351:
6347:
6343:
6339:
6331:
6329:
6328:
6323:
6306:
6305:
6278:
6274:
6267:
6258:
6252:
6243:
6237:
6228:
6202:
6201:
6189:
6188:
6176:
6175:
6150:
6139:
6128:
6121:
6117:
6104:
6102:
6101:
6096:
6091:
6087:
6080:
6071:
6065:
6056:
6050:
6041:
6023:
6014:
6008:
5999:
5968:
5956:rational numbers
5950:
5948:
5947:
5944:
5941:
5934:
5932:
5931:
5926:
5903:
5902:
5869:
5868:
5847:
5838:
5828:
5826:
5825:
5822:
5819:
5811:rational numbers
5808:
5798:
5796:
5795:
5790:
5782:
5741:
5736:
5735:
5710:
5699:
5690:
5671:
5669:
5668:
5663:
5661:
5621:
5615:
5601:
5592:
5583:
5572:
5561:
5559:
5558:
5553:
5551:
5532:
5531:
5506:
5505:
5477:
5476:
5451:
5450:
5421:
5420:
5400:
5398:
5397:
5392:
5390:
5389:
5379:
5361:
5360:
5344:
5340:
5326:
5311:
5307:
5303:
5302:
5300:
5299:
5296:
5293:
5279:dyadic fractions
5276:
5265:
5249:
5247:
5245:
5244:
5241:
5238:
5231:
5229:
5228:
5225:
5222:
5215:
5213:
5212:
5209:
5206:
5199:
5197:
5196:
5193:
5190:
5183:
5181:
5180:
5177:
5174:
5167:
5165:
5164:
5161:
5158:
5151:
5149:
5148:
5145:
5142:
5135:
5133:
5132:
5129:
5126:
5119:
5117:
5116:
5113:
5110:
5103:
5101:
5100:
5097:
5094:
5087:
5085:
5084:
5081:
5078:
5071:
5069:
5068:
5065:
5062:
5046:
5044:
5042:
5041:
5038:
5035:
5028:
5026:
5025:
5022:
5019:
5012:
5010:
5009:
5006:
5003:
4996:
4994:
4993:
4990:
4987:
4980:
4978:
4977:
4974:
4971:
4964:
4962:
4961:
4958:
4955:
4948:
4946:
4945:
4942:
4939:
4932:
4930:
4929:
4926:
4923:
4907:
4905:
4903:
4902:
4899:
4896:
4889:
4887:
4886:
4883:
4880:
4864:
4853:
4836:
4832:
4821:
4817:
4813:
4809:
4805:
4791:
4787:
4780:
4776:
4760:
4759:
4757:
4756:
4751:
4748:
4740:
4731:
4729:
4728:
4723:
4721:
4717:
4715:
4704:
4692:
4684:
4675:
4674:
4672:
4671:
4666:
4663:
4655:
4649:
4647:
4646:
4641:
4639:
4635:
4628:
4620:
4615:
4607:
4604:
4600:
4592:
4584:
4579:
4571:
4551:
4543:
4534:
4532:
4531:
4526:
4524:
4516:
4511:
4506:
4505:
4501:
4493:
4465:
4456:
4454:
4453:
4448:
4446:
4438:
4433:
4428:
4427:
4423:
4415:
4387:
4378:
4377:
4375:
4374:
4371:
4368:
4361:
4359:
4358:
4355:
4352:
4344:
4342:
4341:
4338:
4335:
4328:
4321:
4320:
4318:
4317:
4312:
4309:
4301:
4300:
4298:
4297:
4292:
4289:
4281:
4277:
4266:
4256:. Only positive
4255:
4249:
4247:
4246:
4241:
4239:
4235:
4234:
4232:
4231:
4222:
4221:
4220:
4215:
4211:
4203:
4187:
4186:
4167:
4162:
4160:
4159:
4150:
4149:
4148:
4143:
4139:
4131:
4115:
4114:
4095:
4091:
4087:
4084:
4082:
4081:
4072:
4071:
4070:
4065:
4061:
4053:
4046:
4042:
4035:
4034:
4013:
4008:
4006:
4005:
3996:
3995:
3994:
3989:
3985:
3977:
3961:
3960:
3941:
3921:
3913:
3900:
3898:
3897:
3892:
3890:
3882:
3871:
3863:
3843:
3842:
3840:
3839:
3836:
3833:
3826:
3824:
3823:
3820:
3817:
3810:
3808:
3807:
3804:
3801:
3794:
3792:
3791:
3788:
3785:
3774:
3772:
3771:
3768:
3765:
3758:
3756:
3755:
3752:
3749:
3739:
3737:
3736:
3731:
3729:
3728:
3712:
3710:
3709:
3704:
3702:
3701:
3685:
3683:
3682:
3677:
3675:
3671:
3670:
3669:
3657:
3647:
3643:
3642:
3630:
3619:
3611:
3600:
3583:
3563:
3559:
3555:
3553:
3552:
3547:
3545:
3544:
3535:
3534:
3522:
3521:
3503:
3502:
3486:
3484:
3483:
3478:
3476:
3472:
3468:
3467:
3466:
3457:
3456:
3441:
3440:
3428:
3427:
3412:
3411:
3402:
3401:
3389:
3388:
3370:
3369:
3357:
3356:
3347:
3346:
3334:
3333:
3315:
3314:
3302:
3301:
3292:
3291:
3279:
3278:
3260:
3259:
3239:
3232:
3231:
3219:
3218:
3203:
3202:
3190:
3189:
3142:
3140:
3139:
3134:
3125:
3124:
3100:
3099:
3087:
3086:
3062:
3061:
3043:
3042:
3027:
3026:
3005:
3004:
2992:
2991:
2948:
2946:
2944:
2943:
2940:
2937:
2930:
2928:
2927:
2924:
2921:
2914:
2912:
2911:
2908:
2905:
2898:
2896:
2895:
2892:
2889:
2878:
2876:
2875:
2870:
2859:
2858:
2846:
2845:
2827:
2826:
2808:
2807:
2777:
2772:
2749:
2747:
2746:
2741:
2730:
2729:
2717:
2716:
2692:
2691:
2673:
2672:
2654:
2649:
2620:
2618:
2617:
2612:
2601:
2596:
2585:
2580:
2569:
2564:
2537:
2535:
2534:
2529:
2518:
2507:
2465:
2448:
2416:
2414:
2413:
2408:
2400:
2399:
2375:
2374:
2362:
2361:
2337:
2336:
2318:
2317:
2305:
2304:
2286:
2285:
2273:
2272:
2234:
2232:
2231:
2226:
2215:
2210:
2199:
2194:
2170:
2166:
2155:
2144:
2134:
2132:
2131:
2126:
2081:
2077:
2073:
2071:
2070:
2065:
2057:
2056:
2041:
2040:
2019:
2018:
2006:
2005:
1974:
1941:
1916:
1880:
1876:
1872:
1861:
1857:
1853:
1849:
1845:
1842:with this least
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1794:
1784:
1780:
1748:
1746:
1745:
1742:
1739:
1732:
1730:
1729:
1726:
1723:
1703:
1699:
1692:
1690:
1689:
1686:
1683:
1676:
1674:
1673:
1670:
1667:
1652:
1648:
1644:
1637:
1629:
1614:
1609:
1604:
1599:
1594:
1589:
1584:
1570:
1559:
1556:from generation
1551:
1540:
1525:
1521:
1509:
1498:
1487:
1476:
1465:
1454:
1437:
1426:
1415:
1408:
1399:
1397:
1396:
1391:
1389:
1388:
1378:
1356:
1349:
1334:
1332:
1331:
1326:
1324:
1323:
1313:
1291:
1280:
1270:
1266:
1242:dyadic fractions
1238:finite induction
1215:
1211:
1207:
1192:
1169:
1165:
1161:
1146:
1122:
1112:
1087:
1051:
1047:
1032:
1028:
1024:
1014:
1004:
982:
972:
962:
958:
950:Equivalence rule
925:
921:
917:
913:
905:
879:
875:
868:
864:
860:
848:
844:
799:
792:
784:
777:
776:, ... } = ε
775:
773:
772:
769:
766:
759:
757:
756:
753:
750:
743:
741:
740:
737:
734:
722:
705:
701:
690:
686:
675:
653:
642:dyadic rationals
636:
635:
633:
632:
629:
626:
619:
617:
616:
613:
610:
598:
597:
595:
594:
591:
588:
581:
579:
578:
575:
572:
560:
559:
557:
556:
553:
550:
534:
527:
520:
510:
503:
496:
489:
485:
481:
474:
458:
456:
455:
452:
449:
442:
440:
439:
436:
433:
425:rational numbers
422:
412:
402:
390:
375:
371:
367:
356:, then the pair
355:
351:
347:
343:
339:
335:
331:
321:
311:
307:
295:
277:
258:
246:
234:
230:
197:
192:
186:
126:Research on the
14657:
14656:
14652:
14651:
14650:
14648:
14647:
14646:
14602:
14601:
14600:
14595:
14591:Cours d'Analyse
14569:
14533:
14524:Microcontinuity
14509:Hyperfinite set
14462:
14458:Surreal numbers
14431:
14385:
14376:
14348:Integral symbol
14326:
14321:
14291:
14286:
14263:
14242:
14232:
14205:
14196:Surreal numbers
14186:Ordinal numbers
14131:
14077:
14075:
14072:
14071:
14033:
14000:
13998:
13995:
13994:
13992:
13990:Split-octonions
13961:
13959:
13956:
13955:
13947:
13941:
13923:
13921:
13918:
13917:
13895:
13893:
13890:
13889:
13867:
13865:
13862:
13861:
13858:Complex numbers
13839:
13837:
13834:
13833:
13812:
13761:
13760:
13758:
13755:
13754:
13727:
13725:
13722:
13721:
13694:
13692:
13689:
13688:
13666:
13664:
13661:
13660:
13638:
13636:
13633:
13632:
13629:Natural numbers
13614:
13608:
13578:
13573:
13532:
13501:
13492:Surreal numbers
13466:Ordinal numbers
13395:Actual infinity
13373:
13337:
13286:
13280:
13274:
13219:
13117:Polly Shulman,
13040:
13038:Further reading
13035:
13026:
13024:
13020:
12989:
12980:
12973:
12966:
12952:
12941:
12933:
12929:
12912:
12903:
12896:
12874:
12865:
12828:Ehrlich, Philip
12824:
12811:
12804:
12788:
12775:
12767:
12765:
12744:
12738:
12734:
12702:
12696:
12693:
12685:
12682:
12678:
12668:
12666:
12661:Knuth, Donald.
12659:
12655:
12647:
12645:
12637:
12630:
12620:
12604:
12597:
12593:
12588:
12587:
12582:
12578:
12573:
12569:
12548:
12544:
12539:
12535:
12505:
12501:
12496:
12473:
12466:
12463:
12447:
12431:
12420:
12411:
12338:
12328:
12301:
12293:
12290:
12289:
12270:
12268:
12265:
12264:
12238:
12228:
12209:
12207:
12204:
12203:
12173:
12171:
12168:
12167:
12141:
12139:
12136:
12135:
12111:
12109:
12106:
12105:
12072:
12061:
12056:
12053:
12052:
12041:
11906:
11814:
11733:
11603:is the number −
11545:
11496:
11450:
11160:For any number
11027:if and only if
11011:if and only if
10987:if and only if
10785:) ∧ α < dom(
10601:
10596:
10588:
10580:disjunctive sum
10569:disjunctive sum
10554:
10429:
10418:
10397:
10387:
10386:are games, and
10374:. However, if
10359:
10349:
10322:
10247:
10241:
10198:
10194:
10175:
10168:
10159:
10144:
10141:
10130:
10123:
10110:
10034:
9997:
9994:
9993:
9991:
9964:
9961:
9960:
9958:
9955:
9945:
9936:
9875:and any finite
9854:
9846:
9839:
9831:
9813:
9797:
9789:
9775:
9765:
9764:Similarly, for
9752:
9748:
9740:
9734:
9726:
9713:
9675:
9666:
9658:
9657:
9649:
9646:
9645:
9595:
9586:Similarly, for
9559:
9553:
9536:
9528:
9527:
9525:
9522:
9521:
9500:
9480:
9473:
9466:
9457:
9416:
9412:
9400:
9396:
9388:
9382:
9378:
9363:
9359:
9347:
9343:
9335:
9329:
9325:
9301:
9297:
9291:
9287:
9269:
9265:
9250:
9246:
9240:
9236:
9218:
9214:
9185:
9182:
9181:
9168:
9156:
9143:
9133:
9120:
9115:
9103:
9091:
9062:
9056:
9052:
9050:
9038:
9020:
9017:
9016:
9013:
9011:Basic induction
8990:
8980:
8961:
8953:
8950:
8949:
8924:
8900:
8876:
8874:
8871:
8870:
8851:
8849:
8846:
8845:
8843:
8819:
8815:
8814:
8810:
8804:
8800:
8790:
8783:
8777:
8774:
8773:
8746:
8744:
8741:
8740:
8736:by extension.)
8718:
8710:
8699:
8688:
8677:
8672:
8669:
8668:
8649:
8647:
8644:
8643:
8620:
8618:
8615:
8614:
8595:
8593:
8590:
8589:
8585:
8565:
8554:
8540:
8538:
8535:
8534:
8516:
8515:
8507:
8506:
8504:
8501:
8500:
8467:
8432:
8412:
8409:
8408:
8381:
8380:
8372:
8371:
8369:
8366:
8365:
8342:
8328:
8325:
8324:
8304:
8301:
8282:
8275:
8264:
8257:
8227:
8219:
8211:
8203:
8179:
8149:
8146:
8143:
8142:
8140:
8082:
8066:
8063:
8058:
8057:
8055:
8048:
8045:
8042:
8041:
8039:
8015:
8000:
7993:
7981:
7978:
7973:
7972:
7970:
7969:
7959:
7956:
7951:
7950:
7948:
7934:
7927:
7924:
7919:
7918:
7916:
7902:
7901:is bigger than
7876:
7865:
7835:
7831:
7819:
7813:
7810:
7809:
7801:
7794:
7768:
7758:
7708:
7691:
7688:
7685:
7684:
7682:
7671:
7668:
7663:
7662:
7660:
7659:
7646:
7643:
7640:
7639:
7637:
7626:
7623:
7620:
7619:
7617:
7606:
7603:
7600:
7599:
7597:
7566:
7562:
7555:
7548:
7533:
7518:
7508:
7497:
7493:
7489:
7486:
7478:
7475:
7466:
7463:
7455:
7443:
7437:
7430:
7427:
7421:
7417:
7413:
7406:
7403:
7400:
7399:
7397:
7390:
7387:
7384:
7383:
7381:
7377:
7373:
7366:
7363:
7360:
7359:
7357:
7350:
7347:
7344:
7343:
7341:
7338:
7330:
7326:
7322:
7318:
7314:
7304:
7300:
7293:
7289:
7282:
7275:
7271:
7267:
7264:
7256:
7252:
7248:
7245:
7237:
7233:
7226:
7218:
7212:
7204:
7200:
7193:
7187:
7183:
7179:
7176:
7170:
7166:
7162:
7149:
7145:
7142:
7134:
7130:
7126:
7118:
7115:
7107:
7092:
7082:
7075:
7068:
7065:
7057:
7054:
7046:
7030:
6983:
6966:
6913:
6910:
6909:
6906:
6900:
6891:
6887:
6883:
6870:
6866:
6862:
6858:
6836:
6833:
6832:
6828:
6824:
6820:
6807:
6803:
6799:
6795:
6773:
6770:
6769:
6765:
6761:
6757:
6750:
6746:
6742:
6738:
6734:
6730:
6726:
6719:
6715:
6706:
6702:
6698:
6694:
6690:
6684:
6680:
6676:
6669:
6665:
6656:
6652:
6648:
6644:
6640:
6634:
6630:
6626:
6619:
6615:
6606:
6602:
6598:
6592:
6584:
6569:
6566:
6564:
6549:
6543:
6529:
6526:
6523:
6522:
6520:
6519:
6513:
6504:
6496:
6490:
6481:
6473:
6470:
6462:
6454:
6443:
6436:
6425:
6407:
6398:
6390:
6380:
6376:
6372:
6368:
6365:
6357:
6353:
6349:
6348:(respectively −
6345:
6341:
6337:
6301:
6297:
6256:
6241:
6226:
6213:
6209:
6197:
6193:
6184:
6180:
6171:
6167:
6156:
6153:
6152:
6149:
6141:
6138:
6130:
6123:
6119:
6116:
6108:
6069:
6054:
6039:
6012:
5997:
5990:
5986:
5978:
5975:
5974:
5967:
5959:
5945:
5942:
5939:
5938:
5936:
5898:
5894:
5864:
5860:
5836:
5834:
5831:
5830:
5823:
5820:
5817:
5816:
5814:
5807:
5799:
5781:
5740:
5731:
5727:
5716:
5713:
5712:
5709:
5701:
5698:
5692:
5689:
5681:
5678:
5654:
5652:
5649:
5648:
5630:
5619:
5607:
5600:
5594:
5591:
5585:
5582:
5574:
5571:
5563:
5549:
5548:
5527:
5523:
5507:
5501:
5497:
5494:
5493:
5472:
5468:
5452:
5446:
5442:
5439:
5438:
5422:
5416:
5412:
5408:
5406:
5403:
5402:
5385:
5381:
5369:
5356:
5352:
5350:
5347:
5346:
5342:
5339:
5331:
5317:
5309:
5305:
5297:
5294:
5289:
5288:
5286:
5285:
5275:
5267:
5263:
5256:
5248:< 3 < 4 }
5242:
5239:
5236:
5235:
5233:
5226:
5223:
5220:
5219:
5217:
5210:
5207:
5204:
5203:
5201:
5194:
5191:
5188:
5187:
5185:
5178:
5175:
5172:
5171:
5169:
5162:
5159:
5156:
5155:
5153:
5146:
5143:
5140:
5139:
5137:
5130:
5127:
5124:
5123:
5121:
5114:
5111:
5108:
5107:
5105:
5098:
5095:
5092:
5091:
5089:
5082:
5079:
5076:
5075:
5073:
5066:
5063:
5060:
5059:
5057:
5055:
5049:
5045:< 2 < 3 }
5039:
5036:
5033:
5032:
5030:
5023:
5020:
5017:
5016:
5014:
5007:
5004:
5001:
5000:
4998:
4991:
4988:
4985:
4984:
4982:
4975:
4972:
4969:
4968:
4966:
4959:
4956:
4953:
4952:
4950:
4943:
4940:
4937:
4936:
4934:
4927:
4924:
4921:
4920:
4918:
4916:
4910:
4906:< 1 < 2 }
4900:
4897:
4894:
4893:
4891:
4884:
4881:
4878:
4877:
4875:
4873:
4867:
4862:
4856:
4851:
4845:
4834:
4830:
4819:
4815:
4811:
4807:
4803:
4789:
4785:
4778:
4774:
4767:
4752:
4749:
4746:
4745:
4743:
4742:
4735:
4708:
4703:
4699:
4683:
4681:
4678:
4677:
4667:
4664:
4661:
4660:
4658:
4657:
4653:
4619:
4606:
4583:
4570:
4563:
4560:
4559:
4555:
4542:
4540:
4537:
4536:
4515:
4492:
4488:
4466:
4464:
4462:
4459:
4458:
4437:
4414:
4410:
4388:
4386:
4384:
4381:
4380:
4372:
4369:
4366:
4365:
4363:
4356:
4353:
4350:
4349:
4347:
4346:
4339:
4336:
4333:
4332:
4330:
4323:
4313:
4310:
4307:
4306:
4304:
4303:
4293:
4290:
4287:
4286:
4284:
4283:
4279:
4276:
4268:
4265:
4257:
4253:
4227:
4223:
4216:
4202:
4198:
4197:
4182:
4178:
4168:
4166:
4155:
4151:
4144:
4130:
4126:
4125:
4110:
4106:
4096:
4094:
4077:
4073:
4066:
4052:
4048:
4047:
4030:
4026:
4025:
4021:
4014:
4012:
4001:
3997:
3990:
3976:
3972:
3971:
3956:
3952:
3942:
3940:
3933:
3930:
3929:
3925:
3912:
3910:
3907:
3906:
3881:
3862:
3860:
3857:
3856:
3851:
3837:
3834:
3831:
3830:
3828:
3821:
3818:
3815:
3814:
3812:
3805:
3802:
3799:
3798:
3796:
3789:
3786:
3783:
3782:
3780:
3779:
3769:
3766:
3763:
3762:
3760:
3753:
3750:
3747:
3746:
3744:
3724:
3720:
3718:
3715:
3714:
3697:
3693:
3691:
3688:
3687:
3665:
3661:
3650:
3638:
3634:
3623:
3612:
3604:
3593:
3576:
3575:
3571:
3569:
3566:
3565:
3561:
3557:
3540:
3536:
3530:
3526:
3517:
3513:
3498:
3494:
3492:
3489:
3488:
3474:
3473:
3462:
3458:
3452:
3448:
3436:
3432:
3423:
3419:
3407:
3403:
3397:
3393:
3384:
3380:
3365:
3361:
3352:
3348:
3342:
3338:
3329:
3325:
3310:
3306:
3297:
3293:
3287:
3283:
3274:
3270:
3255:
3251:
3250:
3246:
3237:
3236:
3227:
3223:
3214:
3210:
3198:
3194:
3185:
3181:
3171:
3161:
3159:
3156:
3155:
3148:
3120:
3116:
3095:
3091:
3082:
3078:
3057:
3053:
3038:
3034:
3022:
3018:
3000:
2996:
2987:
2983:
2966:
2963:
2962:
2959:
2941:
2938:
2935:
2934:
2932:
2925:
2922:
2919:
2918:
2916:
2909:
2906:
2903:
2902:
2900:
2893:
2890:
2887:
2886:
2884:
2883:
2854:
2850:
2841:
2837:
2822:
2818:
2803:
2799:
2776:
2771:
2754:
2751:
2750:
2725:
2721:
2712:
2708:
2687:
2683:
2668:
2664:
2653:
2648:
2625:
2622:
2621:
2600:
2595:
2584:
2579:
2568:
2563:
2546:
2543:
2542:
2511:
2500:
2458:
2441:
2424:
2421:
2420:
2395:
2391:
2370:
2366:
2357:
2353:
2332:
2328:
2313:
2309:
2300:
2296:
2281:
2277:
2268:
2264:
2247:
2244:
2243:
2240:
2214:
2209:
2198:
2193:
2176:
2173:
2172:
2168:
2165:
2157:
2154:
2146:
2142:
2087:
2084:
2083:
2079:
2075:
2052:
2048:
2036:
2032:
2014:
2010:
2001:
1997:
1980:
1977:
1976:
1972:
1963:
1951:
1948:
1939:
1930:
1918:
1914:
1905:
1893:
1887:
1878:
1874:
1871:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1804:
1796:
1786:
1782:
1767:
1764:
1743:
1740:
1737:
1736:
1734:
1727:
1724:
1721:
1720:
1718:
1701:
1697:
1687:
1684:
1681:
1680:
1678:
1671:
1668:
1665:
1664:
1662:
1650:
1646:
1642:
1635:
1628:
1622:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1569:
1561:
1557:
1550:
1542:
1539:
1531:
1530:.) Numbers in
1523:
1520:
1511:
1508:
1500:
1497:
1489:
1488:also appear in
1486:
1478:
1475:
1467:
1464:
1456:
1446:
1435:
1425:
1417:
1413:
1407:
1401:
1384:
1380:
1368:
1362:
1359:
1358:
1351:
1348:
1340:
1319:
1315:
1303:
1297:
1294:
1293:
1290:
1282:
1278:
1268:
1264:
1258:
1254:
1226:
1213:
1209:
1202:
1194:
1191:
1182:
1174:
1167:
1163:
1160:
1148:
1145:
1136:
1128:
1114:
1110:
1101:
1089:
1085:
1076:
1064:
1058:
1049:
1045:
1030:
1026:
1016:
1006:
996:
974:
964:
960:
956:
942:, but of their
923:
919:
915:
911:
895:
886:
877:
873:
866:
862:
850:
846:
842:
827:
806:
794:
787:
782:
770:
767:
764:
763:
761:
754:
751:
748:
747:
745:
738:
735:
732:
731:
729:
727:
717:
703:
700:
692:
688:
685:
677:
673:
664:
655:
651:
630:
627:
624:
623:
621:
614:
611:
608:
607:
605:
603:
592:
589:
586:
585:
583:
576:
573:
570:
569:
567:
565:
554:
551:
548:
547:
545:
543:
532:
525:
518:
508:
501:
494:
487:
483:
479:
464:
453:
450:
447:
446:
444:
437:
434:
431:
430:
428:
414:
404:
392:
380:
373:
369:
357:
353:
349:
345:
341:
337:
333:
323:
313:
309:
305:
302:
293:
286:
279:
276:∪ {0, 1, 2}, ∅)
275:
268:
261:
248:
236:
232:
228:
221:
216:
195:
190:
185:
177:
173:Felix Hausdorff
144:surreal numbers
124:
108:ordinal numbers
99:(including the
37:totally ordered
17:
12:
11:
5:
14655:
14645:
14644:
14639:
14634:
14629:
14624:
14619:
14614:
14597:
14596:
14594:
14593:
14588:
14583:
14577:
14575:
14571:
14570:
14568:
14567:
14565:Leonhard Euler
14562:
14557:
14552:
14547:
14541:
14539:
14538:Mathematicians
14535:
14534:
14532:
14531:
14526:
14521:
14516:
14511:
14506:
14501:
14496:
14491:
14486:
14481:
14476:
14470:
14468:
14464:
14463:
14461:
14460:
14455:
14450:
14445:
14439:
14437:
14436:Formalizations
14433:
14432:
14430:
14429:
14424:
14419:
14414:
14409:
14404:
14399:
14393:
14391:
14387:
14386:
14379:
14377:
14375:
14374:
14369:
14362:
14355:
14350:
14345:
14340:
14334:
14332:
14328:
14327:
14324:Infinitesimals
14320:
14319:
14312:
14305:
14297:
14288:
14287:
14285:
14284:
14274:
14272:Classification
14268:
14265:
14264:
14262:
14261:
14259:Normal numbers
14256:
14251:
14229:
14224:
14219:
14213:
14211:
14207:
14206:
14204:
14203:
14198:
14193:
14188:
14183:
14178:
14173:
14168:
14167:
14166:
14156:
14151:
14145:
14143:
14141:infinitesimals
14133:
14132:
14130:
14129:
14128:
14127:
14122:
14117:
14103:
14098:
14093:
14080:
14065:
14060:
14055:
14050:
14044:
14042:
14035:
14034:
14032:
14031:
14026:
14021:
14016:
14003:
13987:
13982:
13977:
13964:
13951:
13949:
13943:
13942:
13940:
13939:
13926:
13911:
13898:
13883:
13870:
13855:
13842:
13822:
13820:
13814:
13813:
13811:
13810:
13805:
13804:
13803:
13793:
13788:
13783:
13778:
13764:
13748:
13743:
13730:
13715:
13710:
13697:
13682:
13669:
13654:
13641:
13625:
13623:
13616:
13615:
13607:
13606:
13599:
13592:
13584:
13575:
13574:
13572:
13571:
13566:
13561:
13556:
13551:
13546:
13540:
13538:
13537:Mathematicians
13534:
13533:
13531:
13530:
13525:
13520:
13515:
13509:
13507:
13503:
13502:
13500:
13499:
13494:
13489:
13484:
13478:
13473:
13468:
13463:
13458:
13453:
13448:
13443:
13441:Gimel function
13438:
13436:Epsilon number
13433:
13427:
13422:
13417:
13412:
13407:
13402:
13397:
13392:
13387:
13381:
13379:
13375:
13374:
13372:
13371:
13366:
13361:
13356:
13351:
13345:
13343:
13339:
13338:
13336:
13335:
13330:
13325:
13320:
13315:
13310:
13305:
13300:
13294:
13292:
13288:
13287:
13273:
13272:
13265:
13258:
13250:
13244:
13243:
13237:
13231:
13226:
13218:
13217:External links
13215:
13214:
13213:
13185:
13158:
13143:
13128:
13115:
13096:Martin Gardner
13093:
13078:
13063:
13039:
13036:
13034:
13033:
12983:Philip Ehrlich
12971:
12964:
12939:
12927:
12901:
12894:
12863:
12809:
12802:
12773:
12755:(1): 365–386,
12732:
12698:
12689:
12676:
12653:
12628:
12618:
12594:
12592:
12589:
12586:
12585:
12576:
12567:
12565:if one exists.
12542:
12533:
12529:epsilon nought
12498:
12497:
12495:
12492:
12491:
12490:
12485:
12479:
12478:
12462:
12459:
12446:
12443:
12419:
12416:
12410:
12407:
12400:
12399:
12360:
12307:
12304:
12300:
12297:
12276:
12273:
12215:
12212:
12202:be subsets of
12192:
12179:
12176:
12147:
12144:
12129:
12117:
12114:
12085:
12082:
12079:
12075:
12071:
12067:
12064:
12060:
12040:
12037:
11905:
11902:
11894:
11893:
11812:
11674:(0) = +1, and
11639:) = +1, and (−
11593:
11592:
11543:
11449:
11446:
11306:
11305:
11291:
11214:
11213:
11158:
10969:
10968:
10917:
10878:
10773:) = min({ dom(
10763:
10762:
10723:
10705:
10605:sign-expansion
10600:
10597:
10595:
10594:Sign expansion
10592:
10587:
10584:
10565:
10564:
10516:
10515:
10512:
10509:
10506:
10487:
10486:
10483:
10480:
10477:
10474:
10468:
10417:
10414:
10279:
10278:
10259:
10243:Main article:
10240:
10237:
10229:transcendental
10167:
10164:
10140:
10139:Exponentiation
10137:
10136:
10135:
10128:
10121:
10115:
10108:
10102:
10101:
10100:
10067:
10049:
10016:
10013:
10010:
10007:
10004:
10001:
9977:
9974:
9971:
9968:
9954:
9951:
9950:
9949:
9948:
9947:
9941:
9932:
9926:
9916:
9915:
9914:
9888:
9866:
9865:
9864:
9863:
9862:
9850:
9835:
9823:
9810:
9804:
9803:
9802:
9793:
9781:
9762:
9744:
9730:
9718:
9706:
9694:
9691:
9688:
9685:
9681:
9678:
9674:
9669:
9664:
9661:
9656:
9653:
9603:
9602:
9601:
9584:
9566:
9556:
9539:
9534:
9531:
9518:
9499:exp satisfies
9497:
9472:
9469:
9462:
9453:
9436:
9433:
9428:
9425:
9422:
9419:
9415:
9411:
9408:
9403:
9399:
9395:
9391:
9385:
9381:
9377:
9374:
9371:
9366:
9362:
9358:
9355:
9350:
9346:
9342:
9338:
9332:
9328:
9324:
9321:
9318:
9313:
9310:
9307:
9304:
9300:
9294:
9290:
9286:
9283:
9280:
9277:
9272:
9268:
9264:
9261:
9258:
9253:
9249:
9243:
9239:
9235:
9232:
9229:
9226:
9221:
9217:
9213:
9210:
9207:
9204:
9201:
9198:
9195:
9192:
9189:
9176:
9175:
9162:
9150:
9149:
9139:
9109:
9097:
9087:
9068:
9065:
9059:
9055:
9047:
9044:
9041:
9037:
9033:
9030:
9027:
9024:
9012:
9009:
9005:exponentiation
8967:
8964:
8960:
8957:
8947:power function
8923:
8920:
8899:
8896:
8882:
8879:
8857:
8854:
8841:
8822:
8818:
8813:
8807:
8803:
8796:
8793:
8789:
8786:
8782:
8752:
8749:
8724:
8721:
8717:
8714:
8709:
8705:
8702:
8698:
8695:
8692:
8687:
8683:
8680:
8676:
8655:
8652:
8626:
8623:
8601:
8598:
8569:
8564:
8560:
8557:
8553:
8550:
8546:
8543:
8519:
8513:
8510:
8486:
8483:
8480:
8477:
8474:
8470:
8466:
8463:
8460:
8457:
8454:
8451:
8448:
8445:
8442:
8439:
8435:
8431:
8428:
8425:
8422:
8419:
8416:
8384:
8378:
8375:
8348:
8345:
8341:
8338:
8335:
8332:
8300:
8297:
8280:
8273:
8267:
8266:
8262:
8255:
8234:
8233:
8225:
8217:
8209:
8201:
8157:
8156:
8137:
8106:
8105:
8081:
8078:
8004:
8003:
7998:
7991:
7965:is defined by
7860:Note that the
7844:
7841:
7838:
7834:
7828:
7825:
7822:
7818:
7755:
7754:
7705:
7704:
7657:
7530:
7529:
7482:
7474:
7471:
7459:
7425:
7334:
7260:
7241:
7222:
7208:
7191:
7174:
7138:
7122:
7111:
7061:
7050:
7034:
7029:
7026:
7023:
7020:
7017:
7014:
7011:
7008:
7005:
7002:
6999:
6996:
6993:
6990:
6987:
6982:
6979:
6976:
6973:
6970:
6965:
6962:
6959:
6956:
6953:
6950:
6947:
6944:
6941:
6938:
6935:
6932:
6929:
6926:
6923:
6920:
6917:
6904:
6898:
6897:
6896:
6895:
6880:
6846:
6843:
6840:
6817:
6783:
6780:
6777:
6754:
6713:
6688:
6663:
6638:
6613:
6588:
6565:
6560:
6554:
6509:
6500:
6486:
6477:
6466:
6459:
6458:
6452:
6441:
6434:
6423:
6394:
6361:
6321:
6318:
6315:
6312:
6309:
6304:
6300:
6296:
6293:
6290:
6287:
6284:
6281:
6277:
6273:
6270:
6264:
6261:
6255:
6249:
6246:
6240:
6234:
6231:
6225:
6222:
6219:
6216:
6212:
6208:
6205:
6200:
6196:
6192:
6187:
6183:
6179:
6174:
6170:
6166:
6163:
6160:
6145:
6134:
6112:
6094:
6090:
6086:
6083:
6077:
6074:
6068:
6062:
6059:
6053:
6047:
6044:
6038:
6035:
6032:
6029:
6026:
6020:
6017:
6011:
6005:
6002:
5996:
5993:
5989:
5985:
5982:
5963:
5924:
5921:
5918:
5915:
5912:
5909:
5906:
5901:
5897:
5893:
5890:
5887:
5884:
5881:
5878:
5875:
5872:
5867:
5863:
5859:
5856:
5853:
5850:
5844:
5841:
5803:
5788:
5785:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5747:
5744:
5739:
5734:
5730:
5726:
5723:
5720:
5705:
5696:
5685:
5677:
5674:
5660:
5657:
5628:
5625:epsilon number
5598:
5589:
5578:
5567:
5547:
5544:
5541:
5538:
5535:
5530:
5526:
5522:
5519:
5516:
5513:
5510:
5508:
5504:
5500:
5496:
5495:
5492:
5489:
5486:
5483:
5480:
5475:
5471:
5467:
5464:
5461:
5458:
5455:
5453:
5449:
5445:
5441:
5440:
5437:
5434:
5431:
5428:
5425:
5423:
5419:
5415:
5411:
5410:
5388:
5384:
5378:
5375:
5372:
5368:
5364:
5359:
5355:
5335:
5271:
5260:natural number
5255:
5252:
5251:
5250:
5053:
5047:
4933:< −1 < −
4914:
4908:
4871:
4865:
4860:
4854:
4849:
4839:
4838:
4823:
4796:
4793:
4782:
4766:
4763:
4761:is undefined.
4720:
4714:
4711:
4707:
4702:
4698:
4695:
4690:
4687:
4638:
4634:
4631:
4626:
4623:
4618:
4613:
4610:
4603:
4598:
4595:
4590:
4587:
4582:
4577:
4574:
4569:
4566:
4562:
4558:
4554:
4549:
4546:
4522:
4519:
4514:
4509:
4504:
4499:
4496:
4491:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4444:
4441:
4436:
4431:
4426:
4421:
4418:
4413:
4409:
4406:
4403:
4400:
4397:
4394:
4391:
4272:
4261:
4238:
4230:
4226:
4219:
4214:
4209:
4206:
4201:
4196:
4193:
4190:
4185:
4181:
4177:
4174:
4171:
4165:
4158:
4154:
4147:
4142:
4137:
4134:
4129:
4124:
4121:
4118:
4113:
4109:
4105:
4102:
4099:
4090:
4080:
4076:
4069:
4064:
4059:
4056:
4051:
4045:
4041:
4038:
4033:
4029:
4024:
4020:
4017:
4011:
4004:
4000:
3993:
3988:
3983:
3980:
3975:
3970:
3967:
3964:
3959:
3955:
3951:
3948:
3945:
3939:
3936:
3932:
3928:
3924:
3919:
3916:
3888:
3885:
3880:
3877:
3874:
3869:
3866:
3850:
3847:
3846:
3845:
3727:
3723:
3700:
3696:
3674:
3668:
3664:
3660:
3656:
3653:
3646:
3641:
3637:
3633:
3629:
3626:
3622:
3618:
3615:
3610:
3607:
3603:
3599:
3596:
3592:
3589:
3586:
3582:
3579:
3574:
3543:
3539:
3533:
3529:
3525:
3520:
3516:
3512:
3509:
3506:
3501:
3497:
3471:
3465:
3461:
3455:
3451:
3447:
3444:
3439:
3435:
3431:
3426:
3422:
3418:
3415:
3410:
3406:
3400:
3396:
3392:
3387:
3383:
3379:
3376:
3373:
3368:
3364:
3360:
3355:
3351:
3345:
3341:
3337:
3332:
3328:
3324:
3321:
3318:
3313:
3309:
3305:
3300:
3296:
3290:
3286:
3282:
3277:
3273:
3269:
3266:
3263:
3258:
3254:
3249:
3245:
3242:
3240:
3238:
3235:
3230:
3226:
3222:
3217:
3213:
3209:
3206:
3201:
3197:
3193:
3188:
3184:
3180:
3177:
3174:
3172:
3170:
3167:
3164:
3163:
3147:
3146:Multiplication
3144:
3132:
3128:
3123:
3119:
3115:
3112:
3109:
3106:
3103:
3098:
3094:
3090:
3085:
3081:
3077:
3074:
3071:
3068:
3065:
3060:
3056:
3052:
3049:
3046:
3041:
3037:
3033:
3030:
3025:
3021:
3017:
3014:
3011:
3008:
3003:
2999:
2995:
2990:
2986:
2982:
2979:
2976:
2973:
2970:
2958:
2955:
2951:
2950:
2868:
2865:
2862:
2857:
2853:
2849:
2844:
2840:
2836:
2833:
2830:
2825:
2821:
2817:
2814:
2811:
2806:
2802:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2775:
2770:
2767:
2764:
2761:
2758:
2739:
2736:
2733:
2728:
2724:
2720:
2715:
2711:
2707:
2704:
2701:
2698:
2695:
2690:
2686:
2682:
2679:
2676:
2671:
2667:
2663:
2660:
2657:
2652:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2610:
2607:
2604:
2599:
2594:
2591:
2588:
2583:
2578:
2575:
2572:
2567:
2562:
2559:
2556:
2553:
2550:
2527:
2524:
2521:
2517:
2514:
2510:
2506:
2503:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2477:
2474:
2471:
2468:
2464:
2461:
2457:
2454:
2451:
2447:
2444:
2440:
2437:
2434:
2431:
2428:
2406:
2403:
2398:
2394:
2390:
2387:
2384:
2381:
2378:
2373:
2369:
2365:
2360:
2356:
2352:
2349:
2346:
2343:
2340:
2335:
2331:
2327:
2324:
2321:
2316:
2312:
2308:
2303:
2299:
2295:
2292:
2289:
2284:
2280:
2276:
2271:
2267:
2263:
2260:
2257:
2254:
2251:
2239:
2236:
2224:
2221:
2218:
2213:
2208:
2205:
2202:
2197:
2192:
2189:
2186:
2183:
2180:
2161:
2150:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2063:
2060:
2055:
2051:
2047:
2044:
2039:
2035:
2031:
2028:
2025:
2022:
2017:
2013:
2009:
2004:
2000:
1996:
1993:
1990:
1987:
1984:
1975:is defined by
1968:
1959:
1947:
1944:
1935:
1926:
1910:
1901:
1886:
1883:
1877:in generation
1867:
1800:
1763:
1760:
1751:
1750:
1749:< 1 < 2.
1706:complete forms
1658:
1657:
1654:
1632:
1626:
1616:
1615:
1610:
1605:
1600:
1595:
1590:
1585:
1571:is called its
1565:
1546:
1535:
1515:
1504:
1493:
1482:
1471:
1460:
1421:
1405:
1387:
1383:
1377:
1374:
1371:
1367:
1344:
1337:
1336:
1322:
1318:
1312:
1309:
1306:
1302:
1286:
1276:ordinal number
1272:
1262:
1253:
1252:Induction rule
1250:
1225:
1222:
1218:
1217:
1198:
1187:
1178:
1171:
1156:
1141:
1132:
1106:
1097:
1081:
1072:
1057:
1054:
985:
984:
936:surreal number
932:
931:
928:order relation
885:
882:
845:and right set
826:
823:
805:
802:
800:and so forth.
779:
778:
724:
723:
696:
681:
669:
660:
638:
637:
600:
599:
562:
561:
536:
535:
529:
528:
522:
521:
512:
511:
505:
504:
498:
497:
301:
298:
291:
284:
273:
266:
220:
217:
215:
212:
181:
123:
120:
56:absolute value
33:surreal number
15:
9:
6:
4:
3:
2:
14654:
14643:
14640:
14638:
14635:
14633:
14630:
14628:
14625:
14623:
14620:
14618:
14615:
14613:
14610:
14609:
14607:
14592:
14589:
14587:
14584:
14582:
14579:
14578:
14576:
14572:
14566:
14563:
14561:
14558:
14556:
14553:
14551:
14548:
14546:
14543:
14542:
14540:
14536:
14530:
14527:
14525:
14522:
14520:
14517:
14515:
14512:
14510:
14507:
14505:
14502:
14500:
14497:
14495:
14492:
14490:
14487:
14485:
14482:
14480:
14477:
14475:
14472:
14471:
14469:
14465:
14459:
14456:
14454:
14451:
14449:
14446:
14444:
14443:Differentials
14441:
14440:
14438:
14434:
14428:
14425:
14423:
14420:
14418:
14415:
14413:
14410:
14408:
14405:
14403:
14400:
14398:
14395:
14394:
14392:
14388:
14383:
14373:
14370:
14368:
14367:
14363:
14361:
14360:
14356:
14354:
14351:
14349:
14346:
14344:
14341:
14339:
14336:
14335:
14333:
14329:
14325:
14318:
14313:
14311:
14306:
14304:
14299:
14298:
14295:
14283:
14275:
14273:
14270:
14269:
14266:
14260:
14257:
14255:
14252:
14249:
14245:
14239:
14235:
14230:
14228:
14225:
14223:
14222:Fuzzy numbers
14220:
14218:
14215:
14214:
14212:
14208:
14202:
14199:
14197:
14194:
14192:
14189:
14187:
14184:
14182:
14179:
14177:
14174:
14172:
14169:
14165:
14162:
14161:
14160:
14157:
14155:
14152:
14150:
14147:
14146:
14144:
14142:
14138:
14134:
14126:
14123:
14121:
14118:
14116:
14113:
14112:
14111:
14107:
14104:
14102:
14099:
14097:
14094:
14069:
14066:
14064:
14061:
14059:
14056:
14054:
14051:
14049:
14046:
14045:
14043:
14041:
14036:
14030:
14027:
14025:
14024:Biquaternions
14022:
14020:
14017:
13991:
13988:
13986:
13983:
13981:
13978:
13953:
13952:
13950:
13944:
13915:
13912:
13887:
13884:
13859:
13856:
13831:
13827:
13824:
13823:
13821:
13819:
13815:
13809:
13806:
13802:
13799:
13798:
13797:
13794:
13792:
13789:
13787:
13784:
13782:
13779:
13752:
13749:
13747:
13744:
13719:
13716:
13714:
13711:
13686:
13683:
13658:
13655:
13630:
13627:
13626:
13624:
13622:
13617:
13612:
13605:
13600:
13598:
13593:
13591:
13586:
13585:
13582:
13570:
13567:
13565:
13562:
13560:
13557:
13555:
13552:
13550:
13549:David Hilbert
13547:
13545:
13542:
13541:
13539:
13535:
13529:
13526:
13524:
13521:
13519:
13516:
13514:
13511:
13510:
13508:
13504:
13498:
13495:
13493:
13490:
13488:
13485:
13482:
13479:
13477:
13474:
13472:
13469:
13467:
13464:
13462:
13461:Infinitesimal
13459:
13457:
13454:
13452:
13449:
13447:
13446:Hilbert space
13444:
13442:
13439:
13437:
13434:
13431:
13428:
13426:
13423:
13421:
13418:
13416:
13413:
13411:
13408:
13406:
13403:
13401:
13398:
13396:
13393:
13391:
13388:
13386:
13383:
13382:
13380:
13376:
13370:
13367:
13365:
13362:
13360:
13357:
13355:
13352:
13350:
13347:
13346:
13344:
13340:
13334:
13331:
13329:
13326:
13324:
13321:
13319:
13316:
13314:
13311:
13309:
13306:
13304:
13301:
13299:
13296:
13295:
13293:
13289:
13283:
13278:
13271:
13266:
13264:
13259:
13257:
13252:
13251:
13248:
13241:
13238:
13235:
13232:
13230:
13227:
13224:
13221:
13220:
13211:
13205:
13201:
13197:
13193:
13192:
13186:
13183:
13182:0-7456-3878-3
13179:
13176:(paperback),
13175:
13174:0-7456-3879-1
13171:
13167:
13163:
13159:
13156:
13155:0-521-31205-1
13152:
13148:
13144:
13141:
13140:0-444-70226-1
13137:
13133:
13129:
13126:
13125:
13120:
13116:
13113:
13109:
13108:0-7167-1987-8
13105:
13101:
13097:
13094:
13091:
13090:1-56881-130-6
13087:
13083:
13079:
13076:
13075:1-56881-127-6
13072:
13068:
13064:
13061:
13057:
13056:0-201-03812-9
13053:
13049:
13045:
13042:
13041:
13023:on 2017-10-07
13019:
13015:
13011:
13007:
13003:
12999:
12995:
12988:
12984:
12978:
12976:
12967:
12965:0-444-70226-1
12961:
12957:
12950:
12948:
12946:
12944:
12936:
12931:
12922:
12917:
12910:
12908:
12906:
12897:
12895:9780521312059
12891:
12887:
12883:
12879:
12872:
12870:
12868:
12859:
12855:
12850:
12845:
12841:
12837:
12833:
12829:
12822:
12820:
12818:
12816:
12814:
12805:
12803:9781568811277
12799:
12795:
12794:
12786:
12784:
12782:
12780:
12778:
12763:
12758:
12754:
12750:
12743:
12736:
12729:
12725:
12720:
12715:
12711:
12707:
12701:
12692:
12688:
12680:
12664:
12657:
12644:
12643:
12635:
12633:
12625:
12621:
12619:9781461466369
12615:
12611:
12610:
12602:
12600:
12595:
12580:
12571:
12564:
12560:
12556:
12552:
12546:
12537:
12530:
12526:
12522:
12518:
12513:
12509:
12503:
12499:
12489:
12486:
12484:
12481:
12480:
12476:
12470:
12465:
12458:
12456:
12452:
12442:
12438:
12434:
12428:
12425:
12415:
12406:
12403:
12397:
12393:
12389:
12385:
12381:
12377:
12373:
12369:
12365:
12361:
12358:
12354:
12350:
12345:
12341:
12335:
12331:
12326:
12322:
12298:
12295:
12262:
12258:
12254:
12250:
12245:
12241:
12235:
12231:
12201:
12197:
12193:
12165:
12161:
12133:
12130:
12103:
12099:
12098:
12097:
12080:
12077:
12073:
12069:
12049:
12047:
12036:
12034:
12030:
12026:
12022:
12018:
12014:
12010:
12006:
12002:
11998:
11994:
11990:
11986:
11982:
11978:
11974:
11969:
11967:
11963:
11959:
11955:
11951:
11947:
11943:
11939:
11935:
11931:
11927:
11923:
11919:
11915:
11911:
11901:
11899:
11889:
11885:
11881:
11877:
11873:
11869:
11865:
11861:
11857:
11853:
11849:
11845:
11841:
11837:
11833:
11829:
11825:
11821:
11817:
11813:
11808:
11804:
11800:
11796:
11792:
11788:
11784:
11780:
11776:
11772:
11768:
11764:
11760:
11756:
11752:
11748:
11744:
11740:
11736:
11732:
11731:
11730:
11728:
11724:
11720:
11716:
11712:
11708:
11704:
11700:
11696:
11691:
11689:
11685:
11681:
11677:
11673:
11669:
11665:
11661:
11656:
11654:
11650:
11646:
11642:
11638:
11634:
11630:
11626:
11622:
11618:
11614:
11610:
11606:
11602:
11598:
11588:
11584:
11580:
11576:
11572:
11568:
11564:
11560:
11556:
11552:
11548:
11544:
11539:
11535:
11531:
11527:
11523:
11519:
11515:
11511:
11507:
11503:
11499:
11495:
11494:
11493:
11491:
11487:
11483:
11479:
11475:
11471:
11467:
11463:
11459:
11455:
11445:
11443:
11439:
11435:
11431:
11427:
11423:
11419:
11415:
11411:
11407:
11403:
11399:
11395:
11391:
11387:
11383:
11379:
11375:
11371:
11367:
11363:
11359:
11355:
11351:
11347:
11343:
11339:
11333:
11329:
11327:
11323:
11319:
11315:
11311:
11303:
11299:
11295:
11292:
11289:
11285:
11281:
11278:
11277:
11276:
11274:
11270:
11266:
11262:
11258:
11255:For a number
11253:
11251:
11247:
11243:
11239:
11235:
11231:
11227:
11223:
11219:
11216:Furthermore,
11211:
11207:
11203:
11199:
11195:
11191:
11187:
11183:
11179:
11175:
11171:
11167:
11163:
11159:
11156:
11152:
11148:
11144:
11140:
11136:
11132:
11128:
11124:
11123:
11122:
11120:
11116:
11112:
11108:
11104:
11100:
11096:
11092:
11088:
11083:
11081:
11077:
11073:
11069:
11065:
11061:
11057:
11053:
11049:
11045:
11041:
11036:
11034:
11030:
11026:
11022:
11018:
11014:
11010:
11006:
11002:
10998:
10994:
10990:
10986:
10982:
10978:
10974:
10966:
10962:
10958:
10954:
10950:
10946:
10942:
10938:
10934:
10930:
10926:
10922:
10918:
10915:
10911:
10907:
10903:
10899:
10895:
10891:
10887:
10883:
10879:
10876:
10872:
10868:
10864:
10860:
10856:
10852:
10848:
10844:
10840:
10839:
10838:
10836:
10832:
10828:
10824:
10820:
10816:
10812:
10808:
10804:
10800:
10796:
10792:
10788:
10784:
10780:
10776:
10772:
10768:
10760:
10756:
10753:)) = − 1 and
10752:
10748:
10744:
10740:
10736:
10732:
10728:
10724:
10721:
10717:
10713:
10709:
10706:
10703:
10699:
10695:
10691:
10688:
10687:
10686:
10684:
10680:
10675:
10673:
10669:
10665:
10661:
10657:
10653:
10649:
10645:
10644:proper subset
10641:
10637:
10633:
10628:
10626:
10622:
10618:
10614:
10610:
10609:sign-sequence
10606:
10591:
10583:
10581:
10577:
10572:
10570:
10561:
10557:
10552:
10548:
10544:
10543:
10542:
10538:
10536:
10532:
10528:
10524:
10519:
10513:
10510:
10507:
10504:
10503:
10502:
10500:
10495:
10491:
10484:
10481:
10478:
10475:
10472:
10471:Deterministic
10469:
10466:
10462:
10458:
10457:
10456:
10453:
10451:
10447:
10443:
10437:
10433:
10427:
10423:
10413:
10410:
10407:
10403:
10400:
10394:
10390:
10385:
10381:
10377:
10372:
10369:
10365:
10362:
10356:
10352:
10347:
10343:
10339:
10334:
10332:
10326:
10320:
10319:
10314:
10313:
10308:
10307:partial order
10304:
10300:
10295:
10293:
10289:
10282:
10276:
10272:
10268:
10264:
10260:
10257:
10256:
10255:
10253:
10246:
10236:
10234:
10230:
10227:
10223:
10219:
10215:
10211:
10207:
10204:
10192:
10188:
10182:
10178:
10173:
10163:
10155:
10151:
10147:
10134:
10127:
10120:
10116:
10114:
10107:
10103:
10098:
10094:
10090:
10089:
10088:
10084:
10080:
10076:
10072:
10068:
10066:
10062:
10058:
10054:
10050:
10048:
10044:
10040:
10039:
10038:
10032:
10014:
10011:
10005:
9999:
9972:
9966:
9944:
9940:
9935:
9931:
9927:
9924:
9920:
9919:
9917:
9913:
9909:
9905:
9901:
9897:
9893:
9889:
9886:
9882:
9878:
9874:
9870:
9869:
9867:
9859:
9858:
9853:
9849:
9842:
9838:
9834:
9830:
9826:
9821:
9817:
9811:
9808:
9807:
9805:
9800:
9796:
9792:
9788:
9784:
9779:
9772:
9768:
9763:
9760:
9756:
9747:
9743:
9737:
9733:
9729:
9725:
9721:
9716:
9711:
9707:
9692:
9686:
9683:
9667:
9654:
9651:
9643:
9639:
9635:
9632:has the form
9631:
9628:> 0), exp
9627:
9623:
9620:has the form
9619:
9615:
9614:
9612:
9608:
9604:
9598:
9593:
9589:
9585:
9582:
9578:
9577:
9575:
9571:
9567:
9563:
9557:
9537:
9519:
9516:
9512:
9508:
9504:
9498:
9495:
9491:
9488:⇒ 0 < exp
9487:
9483:
9478:
9477:
9476:
9468:
9465:
9461:
9456:
9452:
9447:
9434:
9426:
9423:
9420:
9417:
9409:
9406:
9401:
9397:
9389:
9383:
9379:
9375:
9372:
9369:
9364:
9356:
9353:
9348:
9344:
9336:
9330:
9326:
9322:
9319:
9316:
9311:
9308:
9305:
9302:
9292:
9288:
9284:
9281:
9275:
9270:
9266:
9262:
9259:
9256:
9251:
9241:
9237:
9233:
9230:
9224:
9219:
9215:
9211:
9208:
9205:
9202:
9196:
9193:
9190:
9187:
9179:
9172:
9166:
9160:
9155:
9154:
9153:
9147:
9142:
9137:
9132:
9131:
9130:
9127:
9123:
9117:
9113:
9107:
9101:
9095:
9090:
9085:
9066:
9063:
9057:
9053:
9045:
9042:
9039:
9035:
9031:
9028:
9025:
9022:
9008:
9006:
9001:
8997:
8993:
8987:
8983:
8958:
8955:
8948:
8944:
8939:
8937:
8933:
8929:
8919:
8917:
8914:) (with base
8913:
8909:
8905:
8895:
8840:
8820:
8816:
8811:
8805:
8801:
8787:
8784:
8780:
8771:
8767:
8737:
8715:
8707:
8693:
8685:
8641:
8583:
8562:
8548:
8497:
8481:
8478:
8475:
8472:
8464:
8461:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8429:
8426:
8420:
8417:
8406:
8403:
8401:
8363:
8362:Dedekind cuts
8339:
8336:
8333:
8330:
8322:
8318:
8312:
8308:
8296:
8294:
8290:
8285:
8279:
8272:
8261:
8254:
8250:
8247:
8246:
8245:
8243:
8239:
8231:
8223:
8215:
8207:
8206:
8205:
8199:
8195:
8191:
8186:
8182:
8177:
8173:
8169:
8165:
8160:
8138:
8135:
8134:
8133:
8131:
8127:
8123:
8119:
8115:
8111:
8103:
8099:
8095:
8094:
8093:
8091:
8087:
8077:
8075:
8061:
8037:
8031:
8027:
8023:
8019:
8013:
8009:
7997:
7990:
7976:
7968:
7967:
7966:
7954:
7946:
7941:
7937:
7922:
7914:
7909:
7905:
7900:
7895:
7894:of ordinals.
7893:
7888:
7884:
7880:
7874:
7869:
7863:
7858:
7842:
7839:
7836:
7832:
7826:
7823:
7820:
7816:
7807:
7802:2 = { 1 | }
7797:
7792:
7788:
7784:
7780:
7775:
7771:
7765:
7761:
7752:
7748:
7744:
7740:
7736:
7732:
7728:
7724:
7723:
7722:
7720:
7715:
7711:
7700:
7680:
7666:
7658:
7655:
7635:
7615:
7595:
7591:
7587:
7583:
7579:
7575:
7574:
7573:
7569:
7563:−1 = { | 0 }
7558:
7551:
7546:
7542:
7536:
7525:
7521:
7517:
7516:
7515:
7511:
7504:
7500:
7485:
7481:
7470:
7462:
7458:
7451:
7447:
7440:
7436:
7424:
7337:
7333:
7317:, along with
7311:
7307:
7297:
7286:
7279:
7263:
7259:
7244:
7240:
7231:
7228:has the same
7225:
7221:
7215:
7211:
7207:
7198:
7190:
7173:
7159:
7155:
7152:
7141:
7137:
7125:
7121:
7114:
7110:
7105:
7101:
7100:Dedekind cuts
7096:
7089:
7085:
7079:
7072:
7064:
7060:
7053:
7049:
7027:
7024:
7021:
7018:
7015:
7012:
7009:
7006:
7003:
7000:
6997:
6994:
6988:
6980:
6977:
6971:
6963:
6960:
6957:
6954:
6951:
6948:
6945:
6942:
6939:
6936:
6933:
6930:
6924:
6921:
6918:
6915:
6903:
6881:
6877:
6873:
6844:
6841:
6838:
6818:
6814:
6810:
6781:
6778:
6775:
6755:
6724:
6723:
6714:
6710:
6689:
6672:
6664:
6660:
6639:
6622:
6614:
6609:
6597:
6596:
6595:
6591:
6587:
6580:
6576:
6572:
6563:
6559:
6553:
6546:
6540:
6532:
6516:
6512:
6508:
6503:
6499:
6493:
6489:
6485:
6480:
6476:
6469:
6465:
6451:
6447:
6440:
6433:
6429:
6422:
6418:
6414:
6410:
6406:
6405:
6404:
6401:
6397:
6393:
6387:
6383:
6364:
6360:
6335:
6334:infinitesimal
6316:
6313:
6310:
6307:
6302:
6298:
6294:
6291:
6288:
6285:
6279:
6275:
6271:
6268:
6262:
6259:
6253:
6247:
6244:
6238:
6232:
6229:
6223:
6220:
6217:
6214:
6210:
6206:
6198:
6194:
6190:
6185:
6181:
6177:
6172:
6168:
6161:
6158:
6148:
6144:
6137:
6133:
6127:
6115:
6111:
6105:
6092:
6088:
6084:
6081:
6075:
6072:
6066:
6060:
6057:
6051:
6045:
6042:
6036:
6033:
6030:
6027:
6024:
6018:
6015:
6009:
6003:
6000:
5994:
5991:
5987:
5983:
5980:
5972:
5966:
5962:
5957:
5952:
5922:
5916:
5913:
5910:
5907:
5904:
5899:
5895:
5891:
5888:
5885:
5882:
5879:
5876:
5873:
5870:
5865:
5861:
5857:
5854:
5848:
5842:
5839:
5829:is given by:
5812:
5806:
5802:
5786:
5778:
5775:
5772:
5769:
5766:
5763:
5760:
5757:
5754:
5751:
5745:
5737:
5732:
5728:
5721:
5718:
5708:
5704:
5695:
5688:
5684:
5673:
5646:
5642:
5641:ordered field
5638:
5632:
5626:
5622:
5614:
5610:
5603:
5597:
5588:
5581:
5577:
5570:
5566:
5542:
5539:
5536:
5533:
5528:
5524:
5520:
5517:
5511:
5509:
5502:
5498:
5487:
5484:
5481:
5478:
5473:
5469:
5465:
5462:
5456:
5454:
5447:
5443:
5432:
5426:
5424:
5417:
5413:
5386:
5382:
5376:
5373:
5370:
5366:
5362:
5357:
5353:
5338:
5334:
5328:
5325:
5321:
5315:
5292:
5284:
5280:
5274:
5270:
5261:
5052:
5048:
4913:
4909:
4870:
4866:
4859:
4855:
4848:
4844:
4843:
4842:
4828:
4824:
4801:
4797:
4794:
4783:
4772:
4771:
4770:
4762:
4755:
4738:
4732:
4718:
4712:
4709:
4705:
4700:
4696:
4693:
4688:
4685:
4670:
4652:For negative
4650:
4636:
4632:
4629:
4624:
4621:
4616:
4611:
4608:
4601:
4596:
4593:
4588:
4585:
4580:
4575:
4572:
4567:
4564:
4556:
4552:
4547:
4544:
4520:
4517:
4512:
4507:
4502:
4497:
4494:
4489:
4482:
4479:
4476:
4470:
4467:
4442:
4439:
4434:
4429:
4424:
4419:
4416:
4411:
4404:
4401:
4398:
4392:
4389:
4326:
4316:
4296:
4275:
4271:
4264:
4260:
4252:for positive
4250:
4236:
4228:
4224:
4217:
4212:
4207:
4204:
4199:
4191:
4188:
4183:
4179:
4172:
4169:
4163:
4156:
4152:
4145:
4140:
4135:
4132:
4127:
4119:
4116:
4111:
4107:
4100:
4097:
4088:
4078:
4074:
4067:
4062:
4057:
4054:
4049:
4043:
4039:
4036:
4031:
4027:
4022:
4018:
4015:
4009:
4002:
3998:
3991:
3986:
3981:
3978:
3973:
3965:
3962:
3957:
3953:
3946:
3943:
3937:
3934:
3926:
3922:
3917:
3914:
3904:
3901:
3886:
3883:
3878:
3875:
3872:
3867:
3864:
3854:
3778:
3777:
3776:
3741:
3725:
3721:
3698:
3694:
3672:
3666:
3662:
3658:
3654:
3651:
3644:
3639:
3635:
3631:
3627:
3624:
3620:
3616:
3613:
3608:
3605:
3601:
3597:
3594:
3590:
3587:
3584:
3580:
3577:
3572:
3541:
3537:
3531:
3527:
3523:
3518:
3514:
3510:
3507:
3504:
3499:
3495:
3469:
3463:
3459:
3453:
3449:
3445:
3442:
3437:
3433:
3429:
3424:
3420:
3416:
3413:
3408:
3404:
3398:
3394:
3390:
3385:
3381:
3377:
3374:
3371:
3366:
3362:
3358:
3353:
3349:
3343:
3339:
3335:
3330:
3326:
3322:
3319:
3316:
3311:
3307:
3303:
3298:
3294:
3288:
3284:
3280:
3275:
3271:
3267:
3264:
3261:
3256:
3252:
3247:
3243:
3241:
3228:
3224:
3220:
3215:
3211:
3199:
3195:
3191:
3186:
3182:
3175:
3173:
3168:
3165:
3153:
3143:
3130:
3121:
3117:
3113:
3110:
3107:
3104:
3101:
3096:
3092:
3088:
3083:
3079:
3075:
3072:
3069:
3066:
3063:
3058:
3054:
3047:
3039:
3035:
3031:
3028:
3023:
3019:
3015:
3009:
3001:
2997:
2993:
2988:
2984:
2977:
2974:
2971:
2968:
2954:
2882:
2881:
2880:
2879:For example:
2866:
2863:
2855:
2851:
2847:
2842:
2838:
2831:
2823:
2819:
2815:
2812:
2809:
2804:
2800:
2796:
2793:
2787:
2784:
2781:
2773:
2765:
2762:
2759:
2756:
2737:
2734:
2726:
2722:
2718:
2713:
2709:
2702:
2696:
2693:
2688:
2684:
2680:
2677:
2674:
2669:
2665:
2658:
2650:
2642:
2639:
2636:
2633:
2630:
2627:
2608:
2605:
2597:
2589:
2581:
2573:
2565:
2557:
2554:
2551:
2548:
2539:
2522:
2519:
2515:
2512:
2508:
2504:
2501:
2497:
2494:
2488:
2485:
2482:
2479:
2475:
2469:
2466:
2462:
2459:
2455:
2452:
2449:
2445:
2442:
2435:
2432:
2429:
2426:
2418:
2404:
2396:
2392:
2388:
2385:
2382:
2379:
2376:
2371:
2367:
2363:
2358:
2354:
2350:
2347:
2344:
2341:
2338:
2333:
2329:
2322:
2314:
2310:
2306:
2301:
2297:
2290:
2282:
2278:
2274:
2269:
2265:
2258:
2255:
2252:
2249:
2235:
2222:
2219:
2211:
2203:
2195:
2187:
2184:
2181:
2178:
2164:
2160:
2153:
2149:
2140:
2135:
2122:
2116:
2113:
2110:
2107:
2104:
2101:
2095:
2092:
2089:
2061:
2053:
2049:
2045:
2042:
2037:
2033:
2029:
2023:
2015:
2011:
2007:
2002:
1998:
1991:
1988:
1985:
1982:
1971:
1967:
1962:
1958:
1954:
1943:
1938:
1934:
1929:
1925:
1921:
1913:
1909:
1904:
1900:
1896:
1892:
1882:
1870:
1866:
1803:
1799:
1793:
1789:
1778:
1774:
1770:
1759:
1757:
1716:
1715:
1714:
1711:
1707:
1694:
1655:
1641:
1633:
1625:
1621:
1620:
1619:
1611:
1606:
1601:
1596:
1591:
1586:
1581:
1580:
1579:
1576:
1574:
1568:
1564:
1555:
1549:
1545:
1538:
1534:
1529:
1528:limit ordinal
1518:
1514:
1507:
1503:
1496:
1492:
1485:
1481:
1474:
1470:
1463:
1459:
1453:
1449:
1443:
1441:
1432:
1430:
1424:
1420:
1410:
1404:
1385:
1381:
1375:
1372:
1369:
1365:
1354:
1347:
1343:
1320:
1316:
1310:
1307:
1304:
1300:
1289:
1285:
1277:
1273:
1261:
1256:
1255:
1249:
1247:
1243:
1239:
1235:
1231:
1221:
1206:
1201:
1197:
1190:
1186:
1181:
1177:
1172:
1159:
1155:
1151:
1144:
1140:
1135:
1131:
1126:
1125:
1124:
1121:
1117:
1109:
1105:
1100:
1096:
1092:
1084:
1080:
1075:
1071:
1067:
1061:
1053:
1042:
1040:
1036:
1023:
1019:
1013:
1009:
1003:
999:
994:
993:antisymmetric
990:
981:
977:
971:
967:
954:
953:
952:
951:
947:
945:
941:
937:
929:
909:
903:
899:
893:
892:
891:
890:
881:
874:{ { } | { } }
870:
858:
854:
840:
836:
832:
822:
819:
815:
811:
801:
797:
791:
726:
725:
721:
716:
715:
714:
711:
709:
699:
695:
684:
680:
672:
668:
663:
659:
650:
645:
643:
602:
601:
564:
563:
542:
541:
540:
531:
530:
524:
523:
517:
516:
515:
507:
506:
500:
499:
493:
492:
491:
476:
472:
468:
462:
426:
421:
417:
411:
407:
400:
396:
388:
384:
377:
365:
361:
330:
326:
320:
316:
297:
294:, 0, 1, 2 | }
290:
283:
272:
265:
256:
252:
244:
240:
226:
211:
208:
204:
199:
189:for ordinals
188:
184:
180:
174:
170:
166:
162:
157:
155:
154:
149:
145:
141:
138:'s 1974 book
137:
133:
129:
119:
117:
113:
109:
106:
102:
98:
94:
90:
86:
82:
81:ordered field
77:
75:
74:
70:'s 1974 book
69:
65:
61:
57:
53:
49:
45:
41:
38:
34:
30:
21:
14499:Internal set
14484:Hyperinteger
14457:
14453:Dual numbers
14364:
14357:
14243:
14233:
14195:
14048:Dual numbers
14040:hypercomplex
13830:Real numbers
13544:Georg Cantor
13518:Möbius plane
13491:
13456:Infinite set
13400:Aleph number
13190:
13184:(hardcover).
13165:
13162:Alain Badiou
13146:
13131:
13122:
13111:
13099:
13081:
13066:
13047:
13044:Donald Knuth
13025:. Retrieved
13018:the original
12997:
12993:
12955:
12930:
12877:
12839:
12835:
12792:
12766:, retrieved
12752:
12748:
12735:
12709:
12705:
12699:
12690:
12686:
12679:
12667:. Retrieved
12656:
12646:, retrieved
12641:
12623:
12608:
12579:
12570:
12562:
12558:
12554:
12550:
12545:
12536:
12502:
12448:
12436:
12432:
12429:
12421:
12412:
12404:
12401:
12395:
12391:
12387:
12383:
12379:
12375:
12371:
12367:
12363:
12356:
12352:
12348:
12343:
12339:
12333:
12329:
12324:
12320:
12260:
12256:
12252:
12248:
12243:
12239:
12233:
12229:
12199:
12195:
12163:
12131:
12050:
12042:
12032:
12028:
12024:
12020:
12016:
12012:
12008:
12004:
12000:
11996:
11992:
11988:
11984:
11980:
11976:
11972:
11970:
11965:
11961:
11957:
11953:
11949:
11945:
11941:
11937:
11933:
11929:
11925:
11921:
11917:
11913:
11909:
11907:
11897:
11895:
11887:
11883:
11879:
11875:
11871:
11867:
11863:
11859:
11855:
11851:
11847:
11843:
11839:
11835:
11831:
11827:
11823:
11819:
11815:
11806:
11802:
11798:
11794:
11790:
11786:
11782:
11778:
11774:
11770:
11766:
11762:
11758:
11754:
11750:
11746:
11742:
11738:
11734:
11726:
11722:
11718:
11714:
11710:
11706:
11702:
11698:
11694:
11693:The product
11692:
11687:
11683:
11675:
11671:
11667:
11659:
11657:
11652:
11648:
11644:
11640:
11636:
11632:
11628:
11624:
11620:
11616:
11615:), and, for
11612:
11608:
11604:
11600:
11596:
11594:
11586:
11582:
11578:
11574:
11570:
11566:
11562:
11558:
11554:
11550:
11546:
11537:
11533:
11529:
11525:
11521:
11517:
11513:
11509:
11505:
11501:
11497:
11489:
11485:
11481:
11477:
11473:
11469:
11465:
11461:
11457:
11453:
11451:
11441:
11437:
11433:
11429:
11425:
11421:
11417:
11413:
11409:
11405:
11401:
11397:
11393:
11389:
11385:
11381:
11377:
11373:
11369:
11365:
11361:
11357:
11353:
11349:
11345:
11341:
11337:
11334:
11330:
11325:
11321:
11317:
11313:
11309:
11307:
11301:
11297:
11293:
11287:
11283:
11279:
11272:
11268:
11264:
11260:
11256:
11254:
11249:
11245:
11241:
11237:
11233:
11229:
11225:
11221:
11217:
11215:
11209:
11205:
11201:
11197:
11193:
11189:
11185:
11181:
11177:
11173:
11169:
11165:
11161:
11154:
11150:
11146:
11142:
11138:
11134:
11130:
11126:
11118:
11114:
11110:
11106:
11102:
11098:
11094:
11090:
11086:
11084:
11080:linear order
11071:
11067:
11063:
11059:
11055:
11051:
11047:
11043:
11037:
11032:
11028:
11024:
11020:
11016:
11012:
11008:
11004:
11000:
10996:
10992:
10988:
10984:
10980:
10976:
10972:
10971:For numbers
10970:
10964:
10960:
10956:
10952:
10948:
10944:
10940:
10936:
10932:
10928:
10924:
10920:
10913:
10909:
10905:
10901:
10897:
10893:
10889:
10885:
10881:
10874:
10870:
10866:
10862:
10858:
10854:
10850:
10846:
10842:
10834:
10830:
10826:
10822:
10818:
10814:
10810:
10806:
10802:
10798:
10794:
10790:
10786:
10782:
10778:
10774:
10770:
10766:
10764:
10758:
10754:
10750:
10746:
10742:
10738:
10734:
10730:
10726:
10719:
10715:
10711:
10707:
10701:
10697:
10693:
10689:
10682:
10678:
10676:
10671:
10667:
10663:
10659:
10655:
10651:
10647:
10639:
10635:
10631:
10629:
10608:
10604:
10602:
10589:
10573:
10566:
10559:
10555:
10550:
10546:
10539:
10522:
10520:
10517:
10498:
10496:
10492:
10488:
10464:
10460:
10454:
10441:
10435:
10431:
10425:
10419:
10408:
10405:
10401:
10398:
10392:
10388:
10383:
10379:
10375:
10370:
10367:
10363:
10360:
10354:
10350:
10345:
10341:
10337:
10335:
10327:
10316:
10310:
10291:
10287:
10283:
10280:
10277:} is a game.
10274:
10270:
10266:
10262:
10251:
10248:
10222:proper class
10190:
10186:
10180:
10176:
10171:
10169:
10153:
10149:
10145:
10142:
10132:
10125:
10118:
10112:
10105:
10096:
10092:
10086:
10082:
10078:
10074:
10070:
10064:
10060:
10056:
10052:
10046:
10042:
10030:
9956:
9942:
9938:
9933:
9929:
9907:
9903:
9899:
9895:
9894:and surreal
9891:
9884:
9880:
9876:
9872:
9861:exponential.
9851:
9847:
9840:
9836:
9832:
9828:
9824:
9819:
9815:
9798:
9794:
9790:
9786:
9782:
9777:
9770:
9766:
9758:
9754:
9745:
9741:
9735:
9731:
9727:
9723:
9719:
9714:
9709:
9641:
9637:
9633:
9629:
9625:
9621:
9617:
9610:
9606:
9596:
9591:
9587:
9580:
9569:
9561:
9514:
9510:
9506:
9502:
9493:
9489:
9485:
9481:
9474:
9463:
9459:
9454:
9450:
9448:
9180:
9177:
9170:
9164:
9158:
9151:
9145:
9140:
9135:
9125:
9121:
9118:
9111:
9105:
9099:
9093:
9088:
9083:
9014:
8999:
8995:
8991:
8985:
8981:
8942:
8940:
8935:
8931:
8925:
8915:
8911:
8901:
8838:
8738:
8582:esoteric pun
8498:
8407:
8404:
8320:
8316:
8310:
8306:
8302:
8286:
8277:
8270:
8269:where every
8268:
8259:
8252:
8248:
8241:
8235:
8229:
8221:
8213:
8197:
8193:
8189:
8184:
8180:
8175:
8171:
8167:
8163:
8161:
8158:
8129:
8125:
8121:
8117:
8113:
8109:
8107:
8101:
8097:
8089:
8083:
8073:
8059:
8035:
8029:
8025:
8021:
8017:
8011:
8007:
8005:
7995:
7988:
7974:
7952:
7947:. That is,
7944:
7939:
7935:
7920:
7912:
7907:
7903:
7898:
7896:
7886:
7882:
7878:
7872:
7867:
7862:conventional
7861:
7859:
7805:
7795:
7790:
7786:
7782:
7778:
7773:
7769:
7763:
7759:
7756:
7750:
7746:
7742:
7738:
7734:
7730:
7726:
7718:
7713:
7709:
7706:
7698:
7678:
7664:
7653:
7633:
7613:
7593:
7589:
7585:
7581:
7577:
7567:
7556:
7549:
7544:
7540:
7534:
7531:
7523:
7519:
7509:
7502:
7498:
7483:
7479:
7476:
7460:
7456:
7449:
7445:
7438:
7422:
7335:
7331:
7309:
7305:
7295:
7284:
7277:
7261:
7257:
7242:
7238:
7223:
7219:
7216:
7209:
7205:
7196:
7188:
7171:
7157:
7153:
7150:
7139:
7135:
7123:
7119:
7112:
7108:
7097:
7087:
7083:
7077:
7070:
7062:
7058:
7051:
7047:
6901:
6899:
6875:
6871:
6812:
6808:
6708:
6670:
6658:
6620:
6607:
6589:
6585:
6578:
6574:
6570:
6567:
6561:
6557:
6556:Contents of
6544:
6538:
6530:
6514:
6510:
6506:
6501:
6497:
6491:
6487:
6483:
6478:
6474:
6467:
6463:
6460:
6449:
6445:
6438:
6431:
6427:
6420:
6416:
6412:
6408:
6402:
6395:
6391:
6385:
6381:
6362:
6358:
6146:
6142:
6135:
6131:
6125:
6113:
6109:
6106:
5971:real numbers
5964:
5960:
5953:
5804:
5800:
5706:
5702:
5693:
5686:
5682:
5679:
5637:proper class
5633:
5612:
5608:
5604:
5595:
5586:
5579:
5575:
5568:
5564:
5336:
5332:
5329:
5323:
5319:
5290:
5272:
5268:
5257:
5232:< 2 <
5184:< 1 <
5072:< 0 <
5050:
5029:< 1 <
4981:< 0 <
4911:
4890:< 0 <
4868:
4857:
4846:
4840:
4799:
4768:
4753:
4736:
4733:
4676:is given by
4668:
4651:
4351:1 + (2 − 3)0
4324:
4314:
4294:
4273:
4269:
4262:
4258:
4251:
3905:
3902:
3855:
3852:
3742:
3149:
2960:
2952:
2540:
2419:
2241:
2162:
2158:
2151:
2147:
2138:
2136:
1969:
1965:
1960:
1956:
1952:
1949:
1936:
1932:
1927:
1923:
1919:
1911:
1907:
1902:
1898:
1894:
1890:
1888:
1868:
1864:
1801:
1797:
1791:
1787:
1776:
1772:
1768:
1765:
1755:
1752:
1733:< 0 <
1709:
1705:
1695:
1659:
1639:
1623:
1617:
1577:
1572:
1566:
1562:
1553:
1547:
1543:
1536:
1532:
1516:
1512:
1505:
1501:
1494:
1490:
1483:
1479:
1472:
1468:
1461:
1457:
1451:
1447:
1444:
1433:
1429:well-ordered
1422:
1418:
1411:
1402:
1352:
1345:
1341:
1338:
1287:
1283:
1259:
1237:
1227:
1219:
1204:
1199:
1195:
1188:
1184:
1179:
1175:
1173:There is no
1157:
1153:
1149:
1142:
1138:
1133:
1129:
1127:There is no
1119:
1115:
1107:
1103:
1098:
1094:
1090:
1082:
1078:
1073:
1069:
1065:
1062:
1059:
1043:
1038:
1034:
1021:
1017:
1011:
1007:
1001:
997:
986:
979:
975:
969:
965:
949:
948:
943:
939:
935:
933:
907:
901:
897:
888:
887:
871:
856:
852:
838:
834:
830:
828:
807:
804:Construction
795:
789:
780:
719:
712:
708:Dedekind cut
697:
693:
682:
678:
670:
666:
661:
657:
646:
639:
544:{ 0 | 1 } =
537:
513:
477:
470:
466:
419:
415:
409:
405:
398:
394:
386:
382:
378:
363:
359:
328:
324:
318:
314:
303:
288:
281:
270:
263:
254:
250:
242:
238:
225:ordered pair
222:
200:
182:
178:
158:
151:
147:
143:
139:
136:Donald Knuth
125:
78:
72:
71:
68:Donald Knuth
44:real numbers
40:proper class
35:system is a
32:
26:
14359:The Analyst
14210:Other types
14029:Bioctonions
13886:Quaternions
13405:Beth number
13062:(archived).
13000:(1): 1–45.
12921:1307.7392v3
12712:: 341–352,
12424:Hahn series
12418:Hahn series
12102:total order
11164:such that ∀
11093:such that ∀
10939:) < dom(
10927:) < dom(
10888:) < dom(
10861:) < dom(
10658:) < dom(
10599:Definitions
10576:Go endgames
10303:total order
10233:isomorphism
9887:is infinite
8928:powers of ω
8586:α = { α | }
8293:Hahn series
8086:archimedean
7753:+4, ... | }
7656:, ... } and
7325:). To map
7321:itself, to
7288:(including
7274:(including
7230:cardinality
5341:for finite
4835:1 = { 0 | }
4765:Consistency
4327:= 3 = { 2 |
2957:Subtraction
849:is written
649:real number
509:{ 2 | } = 3
502:{ 1 | } = 2
495:{ 0 | } = 1
214:Description
165:Hahn series
163:introduced
105:transfinite
29:mathematics
14606:Categories
14338:Adequality
14164:Projective
14137:Infinities
13506:Geometries
13364:Set theory
13027:2017-06-08
12768:2019-03-05
12665:. Stanford
12648:2008-01-24
12591:References
12451:hyperreals
12374:) for all
12319:such that
12100:< is a
11709:) and dom(
11690:(0) = −1.
11647:) = +1 if
11631:) = −1 if
11472:) and dom(
11121:such that
11076:trichotomy
11040:transitive
10955:)) = −1 ∧
10729:such that
10623:and whose
10331:fuzzy game
10210:isomorphic
9739:where all
7435:Cantor set
7148:such that
6568:Given any
5958:appear in
5618:powers of
1885:Arithmetic
1640:other than
1445:For every
1274:Given any
1193:such that
1147:such that
128:Go endgame
116:isomorphic
60:Go endgame
14574:Textbooks
14519:Overspill
14248:solenoids
14068:Sedenions
13914:Octonions
13487:Supertask
12858:0016-2736
12299:∈
12084:⟩
12059:⟨
12051:A triple
12023:) :
11999:) :
11987:}, where
11960:) :
11940:) :
11928:), where
11729:), where
11619:< dom(
11492:), where
11428:) = − or
10077:) = exp (
9693:β
9690:↦
9687:α
9673:→
9554:log = exp
9492:< exp
9407:−
9376:
9354:−
9323:
9317:∣
9285:−
9276:⋅
9263:
9234:−
9225:⋅
9212:
9191:
9169:< exp
9144:< exp
9129:are then
9043:≥
9036:∑
9026:
8959:∈
8821:α
8812:ω
8806:α
8788:∈
8785:α
8781:∑
8766:open sets
8708:∣
8686:∣
8563:∣
8465:∈
8459:∀
8450:∣
8430:∈
8424:∃
8334:∪
8096:ω = { 0,
7897:Just as 2
7885:+ 1 >
7837:ω
7827:ω
7817:⋃
7414:{ 0 } = 1
7378:{ | } = 0
7161:for some
7028:∣
7025:ω
7019:…
6981:∣
6964:∣
6916:ω
6842:∈
6779:∈
6303:∗
6295:∈
6289:∣
6272:…
6218:∣
6191:∣
6178:∪
6173:−
6159:ε
6085:…
6031:∣
6028:…
5981:π
5900:∗
5892:∈
5886:∣
5866:∗
5858:∈
5779:∣
5776:…
5738:∣
5733:∗
5719:ω
5562:of which
5529:∗
5521:∈
5503:−
5474:∗
5466:∈
5374:∈
5367:⋃
5358:∗
5258:For each
4831:0 = { | }
4710:−
4697:−
4633:…
4597:…
4480:−
4402:−
4189:−
4117:−
4037:−
3963:−
3879:⋅
3659:∈
3632:∈
3602:−
3524:−
3446:−
3391:−
3359:∣
3336:−
3281:−
3221:∣
3192:∣
3114:−
3102:−
3089:∣
3076:−
3064:−
3032:−
3029:∣
3016:−
2994:∣
2972:−
2848:∣
2810:∣
2774:∣
2719:∣
2681:∣
2651:∣
2598:∣
2582:∣
2566:∣
2520:∈
2467:∈
2364:∣
2307:∣
2275:∣
2212:∣
2196:∣
2188:−
2179:−
2114:∈
2102:−
2090:−
2046:−
2043:∣
2030:−
2008:∣
1992:−
1983:−
1554:inherited
1366:⋃
1301:⋃
1230:recursive
1224:Induction
839:right set
798:− 1
728:{ 0 | 1,
161:Hans Hahn
46:but also
14622:Infinity
13657:Integers
13619:Sets of
13385:0.999...
13277:Infinity
13149:, 1986,
13134:, 1987,
13124:Discover
13114:article.
13050:, 1974,
13014:18683932
12985:(2012).
12461:See also
12337:and all
12011:) } and
11866: :
11854:) } ∪ {
11830: :
11785: :
11773:) } ∪ {
11749: :
11680:negative
11664:positive
11655:) = −1.
11611:) = dom(
11577: :
11569:) } ∪ {
11557: :
11528: :
11520:) } ∪ {
11508: :
11452:The sum
10967:)) = +1.
10916:)) = −1;
10900:) = dom(
10877:)) = +1;
10849:) = dom(
10817:) = dom(
10813:) = dom(
10761:)) = +1.
10722:)) = −1;
10704:)) = +1;
10625:codomain
10613:function
10535:star (*)
10523:confused
10323:{1 | −1}
10185:, where
10059:and log
9953:Examples
9509:) = exp
8667:we have
8265:ω + ...,
8178:so that
8164:simplest
8136:ω ω = ω,
8024: :
7697:= { 0 |
7217:The set
5676:Infinity
5314:integers
5304:, where
3849:Division
3655:′
3628:′
3617:′
3609:′
3598:′
3581:′
2516:′
2505:′
2463:′
2446:′
2238:Addition
1946:Negation
1573:birthday
1240:are the
1005:(i. e.,
991:must be
861:. When
837:and its
835:left set
620:| 1 } =
420:R′
410:L′
399:R′
395:L′
227:of sets
219:Notation
48:infinite
14642:Numbers
14331:History
14238:numbers
14070: (
13916: (
13888: (
13860: (
13832: (
13753: (
13751:Periods
13720: (
13687: (
13659: (
13631: (
13613:systems
13303:Apeiron
13291:History
13204:3204653
12728:0146089
12386:, then
11920:}) = σ(
11308:then σ(
11019:. Also
10777:), dom(
10654:if dom(
10621:ordinal
10358:, then
10212:to the
10027:
9992:
9988:
9959:
9906:) >
9822:·(1 + Σ
9751:, then
9471:Results
8904:Kruskal
8640:cofinal
8224:) + log
8216:) = log
8153:
8141:
8070:
8056:
8052:
8040:
7985:
7971:
7963:
7949:
7931:
7917:
7871:equals
7695:
7683:
7675:
7661:
7650:
7638:
7630:
7618:
7610:
7598:
7501:+1 = {
7410:
7398:
7394:
7382:
7370:
7358:
7354:
7342:
6869:equals
6806:equals
6705:equals
6655:equals
6535:
6521:
6340:. The
5949:
5937:
5827:
5815:
5680:Define
5616:(using
5593:), but
5301:
5287:
5246:
5234:
5230:
5218:
5214:
5202:
5198:
5186:
5182:
5170:
5166:
5154:
5150:
5138:
5134:
5122:
5118:
5106:
5102:
5090:
5086:
5074:
5070:
5058:
5043:
5031:
5027:
5015:
5011:
4999:
4995:
4983:
4979:
4967:
4963:
4951:
4947:
4935:
4931:
4919:
4904:
4892:
4888:
4876:
4852:= { 0 }
4800:numbers
4758:
4744:
4741:, then
4673:
4659:
4376:
4364:
4360:
4348:
4343:
4331:
4319:
4305:
4299:
4285:
3841:
3829:
3825:
3813:
3809:
3797:
3793:
3781:
3773:
3761:
3757:
3745:
2945:
2933:
2929:
2917:
2913:
2901:
2897:
2885:
2139:numbers
1766:A form
1747:
1735:
1731:
1719:
1691:
1679:
1675:
1663:
1265:= { 0 }
1039:numbers
908:numeric
894:A form
774:
762:
758:
746:
742:
730:
634:
622:
618:
606:
596:
584:
580:
568:
558:
546:
457:
445:
441:
429:
148:numbers
14038:Other
13611:Number
13202:
13180:
13172:
13153:
13138:
13106:
13088:
13073:
13054:
13012:
12962:
12892:
12856:
12800:
12726:
12669:25 May
12616:
12439:) = −1
11979:) = {
11948:} and
11686:) and
11670:) and
11372:∈ dom
11356:∈ dom
11348:∈ dom
11340:∈ dom
11300:) = {
11286:) = {
11003:, and
10931:) ∧ δ(
10892:) ∧ δ(
10853:) ∧ δ(
10793:(α) ≠
10666:(α) =
10662:) and
10619:is an
10617:domain
10615:whose
10382:, and
10344:, and
10152:· log
10148:= exp(
10073:· log
9902:, exp(
9879:, exp(
9855:< 0
9845:, for
9749:> 0
9636:where
9609:, exp
9513:· exp
8323:, and
8108:where
7994:| ω −
7592:| 1 +
4965:< −
4949:< −
3903:where
3648:
2417:where
1355:< 0
781:where
676:where
566:{ 0 |
171:, and
95:, the
91:, the
31:, the
14494:Monad
14246:-adic
14236:-adic
13993:Over
13954:Over
13948:types
13946:Split
13021:(PDF)
13010:S2CID
12990:(PDF)
12916:arXiv
12745:(PDF)
12512:field
12494:Notes
12355:<
12351:<
12251:<
12104:over
12035:) }.
11713:) by
11623:), (−
11476:) by
11416:<
11324:)) =
11275:) by
11192:<
11180:) ∧ ∀
11176:<
11153:<
11141:) ∧ ∀
11137:<
11113:<
11070:>
11054:<
11015:<
11007:>
10991:<
10833:<
10757:(dom(
10749:(dom(
10718:(dom(
10700:(dom(
10681:<
10642:is a
10527:fuzzy
10465:Right
10446:Chess
10318:fuzzy
10315:, or
10299:field
10252:games
10239:Games
10220:by a
10206:field
10069:exp (
9898:>
9579:When
9484:<
9124:<
8941:When
8837:with
8120:<
7376:onto
7372:) of
7329:onto
7266:onto
5322:<
5216:<
5200:<
5168:<
5152:<
5136:<
5120:<
5104:<
5088:<
5013:<
4997:<
4827:field
4822:; and
1891:forms
1790:<
1710:every
1677:and −
1526:is a
1450:<
1436:0 ≤ 0
1350:with
1269:{ | }
1056:Order
1050:{ | }
1046:{ | }
1035:forms
940:forms
878:{ | }
825:Forms
818:pairs
691:and
480:{ | }
187:-sets
14282:List
14139:and
13178:ISBN
13170:ISBN
13151:ISBN
13136:ISBN
13104:ISBN
13086:ISBN
13071:ISBN
13052:ISBN
12960:ISBN
12890:ISBN
12854:ISSN
12798:ISBN
12704:.",
12671:2020
12614:ISBN
12553:and
12394:) ≤
12237:and
12198:and
12194:Let
12160:onto
12074:<
12015:= {
11991:= {
11952:= {
11932:= {
11898:i.e.
11818:= {
11737:= {
11701:and
11597:i.e.
11549:= {
11500:= {
11484:= σ(
11464:and
11380:and
11236:and
11224:and
11089:and
11046:and
10975:and
10943:) ∧
10904:) ∧
10865:) ∧
10825:and
10789:) ∧
10714:and
10696:and
10652:i.e.
10549:and
10463:and
10461:Left
10442:game
10426:Game
10312:zero
10265:and
10189:and
10117:log
10104:exp
10099:here
10085:) =
10051:exp
10041:exp
9928:For
9827:<
9785:<
9776:log
9753:exp
9722:<
9568:For
9501:exp(
9458:and
9157:exp
9152:and
9134:exp
8926:The
8910:exp(
8479:>
8444:<
8258:ω +
8139:ω =
8112:and
8104:ω },
8100:ω |
7987:= {
7877:1 +
7866:1 +
7824:<
7800:and
7785:and
7749:+3,
7745:+2,
7741:+1,
7737:= {
7588:= {
7561:and
7294:1 −
7283:1 −
6827:and
6764:and
6733:and
6718:and
6601:and
6573:= {
6415:= {
6367:are
6314:>
6122:and
6118:are
5914:>
5880:<
5540:<
5485:>
5318:0 ≤
5316:and
5312:are
5308:and
5277:are
4818:and
4810:and
3827:} =
3713:and
3560:and
2156:and
1955:= {
1922:= {
1917:and
1897:= {
1850:and
1818:and
1771:= {
1440:ring
1373:<
1308:<
1093:= {
1088:and
1068:= {
1029:and
1015:and
973:and
959:and
914:and
865:and
831:form
582:} =
514:and
486:and
443:and
413:and
391:and
344:and
336:and
308:and
231:and
50:and
13121:,
13002:doi
12882:doi
12844:doi
12840:167
12757:doi
12753:287
12714:doi
12710:103
12453:in
11968:}.
11912:({
11890:) }
11878:),
11842:),
11809:) }
11797:),
11761:),
11678:is
11662:is
11589:) }
11540:) }
11420:if
11368:if
11252:).
11204:or
11196:),
10959:(δ(
10947:(δ(
10908:(δ(
10869:(δ(
10674:).
10646:of
10638:if
10607:or
10448:or
10336:If
10325:).
10261:If
10224:of
9780:= Σ
9717:= Σ
9708:If
9616:If
9599:– 1
9467:).
9373:exp
9320:exp
9260:exp
9209:exp
9188:exp
9167:+ 1
9023:exp
8894:).
8642:in
8638:is
8208:log
8076:2.
7570:+ 1
7552:− 1
7537:− 1
7505:| }
7454:in
7251:of
7133:of
7102:of
6673:≤ 0
6623:≥ 0
6610:= 0
6583:in
6016:201
5645:set
4739:= 0
4734:If
3759:is
1854:;
1427:is
987:An
946:).
906:is
816:of
812:as
793:or
322:or
130:by
76:.
62:by
27:In
14608::
13828::
13200:MR
13198:.
13164:,
13098:,
13008:.
12998:18
12996:.
12992:.
12974:^
12942:^
12904:^
12888:.
12866:^
12852:.
12838:.
12834:.
12812:^
12776:^
12751:,
12747:,
12724:MR
12722:,
12708:,
12631:^
12622:.
12612:.
12598:^
12457:.
12382:,
12378:∈
12347:,
12342:∈
12332:∈
12259:,
12247:,
12242:∈
12232:∈
12191:).
12027:∈
12003:∈
11983:|
11964:∈
11944:∈
11916:|
11882:∈
11870:∈
11864:uv
11862:−
11860:xv
11858:+
11856:uy
11846:∈
11834:∈
11828:uv
11826:−
11824:xv
11822:+
11820:uy
11801:∈
11789:∈
11783:uv
11781:−
11779:xv
11777:+
11775:uy
11765:∈
11753:∈
11747:uv
11745:−
11743:xv
11741:+
11739:uy
11717:=
11715:xy
11695:xy
11643:)(
11627:)(
11581:∈
11573:+
11561:∈
11553:+
11532:∈
11524:+
11512:∈
11504:+
11480:+
11456:+
11400:∩
11396:=
11388:=
11344:(∀
11328:.
11316:),
11304:},
11290:};
11200:=
11184:∈
11168:∈
11157:),
11145:∈
11129:∈
11105:∈
11097:∈
11066:,
11062:=
11058:,
11035:.
11031:≤
11023:≥
10999:=
10995:∨
10983:≤
10979:,
10919:δ(
10880:δ(
10841:δ(
10829:,
10801:=
10745:,
10737:,
10650:,
10558:+
10537:.
10452:.
10450:Go
10434:|
10430:{
10422:Go
10404:=
10391:=
10378:,
10366:=
10353:=
10340:,
10290:|
10286:{
10273:|
10199:−1
10179:+
10170:A
10131:/
10124:=
10111:=
10081:·
10063:=
10055:=
10045:=
9883:)/
9818:)·
9769:=
9757:=
9161:·
9138:·
9114:+1
9102:+1
8998:·
8994:=
8984:↦
8979:,
8402:.
8309:|
8305:{
8295:.
8251:=
8232:).
8214:xy
8185:rω
8183:−
8028:∈
8020:−
8016:{
8010:−
7938:−
7906:+
7881:=
7772:+
7762:+
7733:+
7729:=
7721::
7712:+
7681:·
7677:=
7652:+
7636:,
7632:+
7616:,
7612:+
7596:,
7584:+
7580:=
7572::
7514::
7512:+1
7469:.
7448:|
7444:{
7396:,
7356:,
7156:=
7095:.
7086:±
6874:−
6811:+
6577:|
6552:.
6537:=
6498:ωS
6475:ωS
6448:·
6444:+
6437:+
6430:·
6426:|
6419:·
6411:·
6400:.
6384:±
6151::
6076:16
6073:51
6058:13
6019:64
6001:25
5711::
5672:.
5611:=
5327:.
4656:,
4589:16
4362:=
3795:⋅
3775::
2931:|
2899:+
2538:.
2223:0.
2082::
1964:|
1931:|
1906:|
1881:.
1775:|
1653:).
1519:−1
1416:,
1248:.
1203:≤
1183:∈
1152:≤
1137:∈
1118:≤
1113:,
1102:|
1077:|
1020:≤
1010:≤
1000:=
978:≤
968:≤
900:|
896:{
880:.
855:|
851:{
829:A
760:,
744:,
674:},
665:|
656:{
604:{
469:|
465:{
418:≠
408:≠
397:|
393:{
385:|
381:{
376:.
362:|
358:{
327:≤
317:≤
312:,
296:.
287:,
280:{
269:∪
253:|
249:{
241:,
156:.
14316:e
14309:t
14302:v
14250:)
14244:p
14240:(
14234:p
14108:/
14092:)
14079:S
14015::
14002:C
13976::
13963:R
13938:)
13925:O
13910:)
13897:H
13882:)
13869:C
13854:)
13841:R
13777:)
13763:P
13742:)
13729:A
13709:)
13696:Q
13681:)
13668:Z
13653:)
13640:N
13603:e
13596:t
13589:v
13285:)
13282:∞
13279:(
13269:e
13262:t
13255:v
13206:.
13157:.
13142:.
13092:.
13077:.
13030:.
13004::
12968:.
12924:.
12918::
12898:.
12884::
12860:.
12846::
12806:.
12759::
12716::
12700:α
12697:ℵ
12691:α
12687:η
12673:.
12563:R
12559:L
12555:R
12551:L
12531:.
12437:ω
12435:(
12433:ν
12396:α
12392:z
12390:(
12388:b
12384:B
12380:A
12376:x
12372:x
12370:(
12368:b
12364:α
12357:y
12353:z
12349:x
12344:B
12340:y
12334:A
12330:x
12325:z
12323:(
12321:b
12306:o
12303:N
12296:z
12275:o
12272:N
12261:B
12257:A
12253:y
12249:x
12244:B
12240:y
12234:A
12230:x
12214:o
12211:N
12200:B
12196:A
12178:o
12175:N
12164:b
12146:o
12143:N
12132:b
12116:o
12113:N
12081:b
12078:,
12070:,
12066:o
12063:N
12033:x
12031:(
12029:R
12025:y
12021:y
12019:(
12017:g
12013:R
12009:x
12007:(
12005:L
12001:y
11997:y
11995:(
11993:g
11989:L
11985:R
11981:L
11977:x
11975:(
11973:g
11966:R
11962:x
11958:x
11956:(
11954:f
11950:S
11946:L
11942:x
11938:x
11936:(
11934:f
11930:M
11926:S
11924:,
11922:M
11918:R
11914:L
11910:f
11892:.
11888:y
11886:(
11884:L
11880:v
11876:x
11874:(
11872:R
11868:u
11852:y
11850:(
11848:R
11844:v
11840:x
11838:(
11836:L
11832:u
11816:R
11811:,
11807:y
11805:(
11803:R
11799:v
11795:x
11793:(
11791:R
11787:u
11771:y
11769:(
11767:L
11763:v
11759:x
11757:(
11755:L
11751:u
11735:L
11727:R
11725:,
11723:L
11721:(
11719:σ
11711:y
11707:x
11703:y
11699:x
11688:x
11684:x
11676:x
11672:x
11668:x
11660:x
11653:α
11651:(
11649:x
11645:α
11641:x
11637:α
11635:(
11633:x
11629:α
11625:x
11621:x
11617:α
11613:x
11609:x
11605:x
11601:x
11591:.
11587:y
11585:(
11583:R
11579:v
11575:v
11571:x
11567:x
11565:(
11563:R
11559:u
11555:y
11551:u
11547:R
11542:,
11538:y
11536:(
11534:L
11530:v
11526:v
11522:x
11518:x
11516:(
11514:L
11510:u
11506:y
11502:u
11498:L
11490:R
11488:,
11486:L
11482:y
11478:x
11474:y
11470:x
11466:y
11462:x
11458:y
11454:x
11442:f
11438:f
11434:z
11432:(
11430:y
11426:z
11424:(
11422:x
11418:y
11414:x
11410:y
11406:x
11402:y
11398:x
11394:z
11390:y
11386:x
11382:y
11378:x
11374:y
11370:x
11366:y
11362:x
11358:f
11354:h
11352:(
11350:g
11346:h
11342:f
11338:g
11336:∀
11326:x
11322:x
11320:(
11318:R
11314:x
11312:(
11310:L
11302:x
11298:x
11296:(
11294:R
11288:x
11284:x
11282:(
11280:L
11273:x
11271:(
11269:R
11265:x
11263:(
11261:L
11257:x
11250:R
11248:,
11246:L
11242:z
11238:R
11234:L
11230:z
11226:R
11222:L
11218:z
11212:.
11210:w
11206:z
11202:z
11198:w
11194:y
11190:w
11188:(
11186:R
11182:y
11178:w
11174:x
11172:(
11170:L
11166:x
11162:w
11155:y
11151:z
11149:(
11147:R
11143:y
11139:z
11135:x
11133:(
11131:L
11127:x
11125:∀
11119:z
11115:y
11111:x
11109:(
11107:R
11103:y
11101:∀
11099:L
11095:x
11091:R
11087:L
11072:y
11068:x
11064:y
11060:x
11056:y
11052:x
11048:y
11044:x
11033:x
11029:y
11025:y
11021:x
11017:x
11013:y
11009:y
11005:x
11001:y
10997:x
10993:y
10989:x
10985:y
10981:x
10977:y
10973:x
10965:y
10963:,
10961:x
10957:y
10953:y
10951:,
10949:x
10945:x
10941:y
10937:y
10935:,
10933:x
10929:x
10925:y
10923:,
10921:x
10914:y
10912:,
10910:x
10906:x
10902:y
10898:y
10896:,
10894:x
10890:x
10886:y
10884:,
10882:x
10875:y
10873:,
10871:x
10867:y
10863:y
10859:y
10857:,
10855:x
10851:x
10847:y
10845:,
10843:x
10835:y
10831:x
10827:y
10823:x
10819:y
10815:x
10811:y
10809:,
10807:x
10803:y
10799:x
10795:y
10791:x
10787:y
10783:x
10779:y
10775:x
10771:y
10769:,
10767:x
10759:z
10755:y
10751:z
10747:x
10743:y
10739:z
10735:x
10731:z
10727:z
10720:y
10716:x
10712:x
10708:y
10702:x
10698:y
10694:y
10690:x
10683:y
10679:x
10672:x
10668:y
10664:x
10660:y
10656:x
10648:y
10640:x
10636:y
10632:x
10563:.
10560:y
10556:x
10551:y
10547:x
10499:x
10467:)
10438:}
10436:R
10432:L
10409:z
10406:y
10402:z
10399:x
10393:y
10389:x
10384:z
10380:y
10376:x
10371:z
10368:y
10364:z
10361:x
10355:y
10351:x
10346:z
10342:y
10338:x
10294:}
10292:0
10288:0
10275:R
10271:L
10267:R
10263:L
10195:i
10191:b
10187:a
10183:i
10181:b
10177:a
10156:)
10154:x
10150:y
10146:x
10133:ω
10129:0
10126:ε
10122:0
10119:ε
10113:ω
10109:0
10106:ε
10097:ω
10093:ω
10087:ω
10083:ω
10079:ω
10075:ω
10071:ω
10065:ω
10061:ω
10057:ω
10053:ω
10047:ω
10043:ω
10035:ω
10033:(
10031:ω
10015:a
10012:=
10009:)
10006:a
10003:(
10000:g
9976:)
9973:a
9970:(
9967:g
9943:β
9939:ε
9934:β
9930:ε
9908:x
9904:x
9900:n
9896:x
9892:n
9885:x
9881:x
9877:n
9873:x
9852:α
9848:a
9843:)
9841:ω
9837:α
9833:s
9829:β
9825:α
9820:r
9816:ω
9814:(
9799:ω
9795:α
9791:r
9787:β
9783:α
9778:x
9771:ω
9767:x
9759:ω
9755:x
9746:α
9742:a
9736:ω
9732:α
9728:r
9724:β
9720:α
9715:x
9710:x
9684::
9680:o
9677:N
9668:+
9663:o
9660:N
9655::
9652:g
9642:α
9638:β
9634:ω
9630:x
9626:α
9624:(
9622:ω
9618:x
9611:x
9607:x
9597:x
9592:x
9588:x
9581:x
9570:x
9565:)
9562:e
9538:+
9533:o
9530:N
9515:y
9511:x
9507:y
9505:+
9503:x
9494:y
9490:x
9486:y
9482:x
9464:R
9460:z
9455:L
9451:z
9435:.
9432:}
9427:1
9424:+
9421:n
9418:2
9414:]
9410:z
9402:L
9398:z
9394:[
9390:/
9384:L
9380:z
9370:,
9365:n
9361:]
9357:z
9349:R
9345:z
9341:[
9337:/
9331:R
9327:z
9312:1
9309:+
9306:n
9303:2
9299:]
9293:R
9289:z
9282:z
9279:[
9271:R
9267:z
9257:,
9252:n
9248:]
9242:L
9238:z
9231:z
9228:[
9220:L
9216:z
9206:,
9203:0
9200:{
9197:=
9194:z
9173:,
9171:x
9165:n
9163:2
9159:y
9146:y
9141:n
9136:x
9126:y
9122:x
9112:n
9110:2
9106:x
9100:n
9098:2
9094:x
9089:n
9084:x
9067:!
9064:n
9058:n
9054:x
9046:0
9040:n
9032:=
9029:x
9000:x
8996:x
8992:x
8986:x
8982:x
8966:o
8963:N
8956:x
8943:y
8936:ω
8932:e
8916:e
8912:x
8881:n
8878:O
8856:o
8853:N
8842:α
8839:a
8817:a
8802:r
8795:o
8792:N
8751:o
8748:N
8723:n
8720:O
8716:=
8713:}
8704:n
8701:O
8697:{
8694:=
8691:}
8682:o
8679:N
8675:{
8654:o
8651:N
8625:n
8622:O
8600:n
8597:O
8568:}
8559:o
8556:N
8552:{
8549:=
8545:n
8542:O
8518:D
8512:o
8509:N
8485:}
8482:n
8476:x
8473::
8469:N
8462:n
8456::
8453:x
8447:n
8441:x
8438::
8434:N
8427:n
8421::
8418:x
8415:{
8383:D
8377:o
8374:N
8347:o
8344:N
8340:=
8337:R
8331:L
8321:R
8317:L
8313:}
8311:R
8307:L
8281:α
8278:y
8274:α
8271:r
8263:1
8260:r
8256:0
8253:r
8249:x
8242:x
8230:y
8228:(
8226:ω
8222:x
8220:(
8218:ω
8212:(
8210:ω
8202:ω
8198:x
8194:y
8190:x
8181:x
8176:y
8172:r
8168:x
8155:,
8150:ω
8147:/
8144:1
8130:r
8126:r
8122:y
8118:x
8114:s
8110:r
8102:s
8098:r
8090:x
8074:ω
8067:2
8064:/
8060:ω
8049:2
8046:/
8043:1
8036:ω
8032:}
8030:Y
8026:y
8022:y
8018:x
8012:Y
8008:x
8001:}
7999:∗
7996:S
7992:∗
7989:S
7982:2
7979:/
7975:ω
7960:2
7957:/
7953:ω
7945:n
7940:n
7936:ω
7928:2
7925:/
7921:ω
7913:n
7908:n
7904:ω
7899:ω
7887:ω
7883:ω
7879:ω
7873:ω
7868:ω
7843:k
7840:+
7833:S
7821:k
7806:ω
7796:ω
7791:ω
7787:ω
7783:ω
7779:ω
7774:ω
7770:ω
7764:ω
7760:ω
7751:ω
7747:ω
7743:ω
7739:ω
7735:ω
7731:ω
7727:ω
7725:2
7719:k
7714:k
7710:ω
7703:.
7701:}
7699:ε
7692:2
7689:/
7686:1
7679:ε
7672:2
7669:/
7665:ε
7654:ε
7647:8
7644:/
7641:1
7634:ε
7627:4
7624:/
7621:1
7614:ε
7607:2
7604:/
7601:1
7594:ε
7590:ε
7586:ε
7582:ε
7578:ε
7576:2
7568:ω
7557:ω
7550:ω
7545:ω
7541:ω
7535:ω
7528:.
7526:}
7524:ω
7520:ω
7510:ω
7503:ω
7499:ω
7494:α
7490:α
7484:ω
7480:S
7467:ω
7461:ω
7457:S
7452:}
7450:R
7446:L
7439:2
7431:I
7426:∗
7423:S
7418:I
7407:9
7404:/
7401:8
7391:9
7388:/
7385:7
7374:I
7367:3
7364:/
7361:2
7351:3
7348:/
7345:1
7336:ω
7332:S
7327:I
7323:y
7319:y
7315:y
7310:ε
7308:±
7306:y
7301:I
7296:ε
7290:ω
7285:ε
7278:ω
7276:−
7272:ε
7268:I
7262:ω
7258:S
7253:R
7249:I
7243:ω
7239:S
7234:R
7224:ω
7220:S
7210:ω
7206:S
7201:Q
7192:∗
7189:S
7184:Q
7180:Q
7175:∗
7172:S
7167:b
7163:a
7158:a
7154:b
7151:x
7146:x
7140:ω
7136:S
7131:Q
7124:ω
7120:S
7113:ω
7109:S
7093:y
7088:ε
7084:y
7078:ε
7076:±
7071:ω
7069:±
7063:ω
7059:S
7052:ω
7048:S
7033:}
7022:,
7016:,
7013:4
7010:,
7007:3
7004:,
7001:2
6998:,
6995:1
6992:{
6989:=
6986:}
6978:0
6975:{
6972:+
6969:}
6961:.
6958:.
6955:.
6952:,
6949:4
6946:,
6943:3
6940:,
6937:2
6934:,
6931:1
6928:{
6925:=
6922:1
6919:+
6905:ω
6902:S
6892:x
6888:L
6884:R
6879:;
6876:ε
6872:y
6867:x
6863:R
6859:L
6845:R
6839:y
6829:R
6825:L
6821:y
6816:;
6813:ε
6809:y
6804:x
6800:R
6796:L
6782:L
6776:y
6766:R
6762:L
6758:y
6753:;
6751:y
6747:x
6743:R
6739:L
6735:R
6731:L
6727:y
6720:R
6716:L
6712:;
6709:ω
6707:−
6703:x
6699:R
6695:n
6691:L
6687:;
6685:n
6681:x
6677:R
6671:n
6666:L
6662:;
6659:ω
6657:+
6653:x
6649:L
6645:n
6641:R
6637:;
6635:n
6631:x
6627:L
6621:n
6616:R
6612:;
6608:x
6603:R
6599:L
6590:ω
6586:S
6581:}
6579:R
6575:L
6571:x
6562:ω
6558:S
6550:ε
6545:ω
6539:ω
6531:ε
6527:/
6524:1
6515:ε
6511:ω
6507:S
6505:·
6502:ω
6492:ε
6488:ω
6484:S
6482:·
6479:ω
6468:ω
6464:S
6457:.
6455:}
6453:∗
6450:S
6446:ε
6442:∗
6439:S
6435:+
6432:S
6428:ω
6424:+
6421:S
6417:ε
6413:ε
6409:ω
6396:ω
6392:S
6386:ε
6382:y
6377:y
6373:ε
6369:ε
6363:ω
6359:S
6354:ω
6350:ε
6346:ε
6342:ω
6338:ε
6320:}
6317:0
6311:y
6308::
6299:S
6292:y
6286:0
6283:{
6280:=
6276:}
6269:,
6263:8
6260:1
6254:,
6248:4
6245:1
6239:,
6233:2
6230:1
6224:,
6221:1
6215:0
6211:{
6207:=
6204:}
6199:+
6195:S
6186:0
6182:S
6169:S
6165:{
6162:=
6147:ω
6143:S
6136:ω
6132:S
6126:ω
6124:−
6120:ω
6114:ω
6110:S
6093:.
6089:}
6082:,
6067:,
6061:4
6052:,
6046:2
6043:7
6037:,
6034:4
6025:,
6010:,
6004:8
5995:,
5992:3
5988:{
5984:=
5965:ω
5961:S
5946:3
5943:/
5940:1
5923:.
5920:}
5917:1
5911:y
5908:3
5905::
5896:S
5889:y
5883:1
5877:y
5874:3
5871::
5862:S
5855:y
5852:{
5849:=
5843:3
5840:1
5824:3
5821:/
5818:1
5805:ω
5801:S
5787:.
5784:}
5773:,
5770:4
5767:,
5764:3
5761:,
5758:2
5755:,
5752:1
5749:{
5746:=
5743:}
5729:S
5725:{
5722:=
5707:ω
5703:S
5697:∗
5694:S
5687:ω
5683:S
5659:o
5656:N
5629:α
5627:ε
5620:ω
5613:ω
5609:β
5599:∗
5596:S
5590:0
5587:S
5580:n
5576:S
5569:∗
5565:S
5546:}
5543:0
5537:x
5534::
5525:S
5518:x
5515:{
5512:=
5499:S
5491:}
5488:0
5482:x
5479::
5470:S
5463:x
5460:{
5457:=
5448:+
5444:S
5436:}
5433:0
5430:{
5427:=
5418:0
5414:S
5387:n
5383:S
5377:N
5371:n
5363:=
5354:S
5343:n
5337:n
5333:S
5324:n
5320:b
5310:b
5306:a
5298:2
5295:/
5291:a
5273:n
5269:S
5264:n
5243:2
5240:/
5237:5
5227:4
5224:/
5221:7
5211:2
5208:/
5205:3
5195:4
5192:/
5189:5
5179:8
5176:/
5173:7
5163:4
5160:/
5157:3
5147:8
5144:/
5141:5
5131:2
5128:/
5125:1
5115:8
5112:/
5109:3
5099:4
5096:/
5093:1
5083:8
5080:/
5077:1
5067:8
5064:/
5061:1
5054:4
5051:S
5040:2
5037:/
5034:3
5024:4
5021:/
5018:3
5008:2
5005:/
5002:1
4992:4
4989:/
4986:1
4976:4
4973:/
4970:1
4960:2
4957:/
4954:1
4944:4
4941:/
4938:3
4928:2
4925:/
4922:3
4915:3
4912:S
4901:2
4898:/
4895:1
4885:2
4882:/
4879:1
4872:2
4869:S
4861:1
4858:S
4850:0
4847:S
4837:.
4820:y
4816:x
4812:y
4808:x
4804:x
4792:;
4790:n
4786:n
4781:;
4779:n
4775:n
4754:y
4750:/
4747:1
4737:y
4719:)
4713:y
4706:1
4701:(
4694:=
4689:y
4686:1
4669:y
4665:/
4662:1
4654:y
4637:}
4630:,
4625:8
4622:3
4617:,
4612:2
4609:1
4602:|
4594:,
4586:5
4581:,
4576:4
4573:1
4568:,
4565:0
4557:{
4553:=
4548:3
4545:1
4521:8
4518:3
4513:=
4508:2
4503:)
4498:4
4495:1
4490:(
4486:)
4483:3
4477:2
4474:(
4471:+
4468:1
4443:4
4440:1
4435:=
4430:2
4425:)
4420:2
4417:1
4412:(
4408:)
4405:3
4399:2
4396:(
4393:+
4390:1
4373:2
4370:/
4367:1
4357:2
4354:/
4340:3
4337:/
4334:1
4325:y
4315:y
4311:/
4308:1
4295:y
4291:/
4288:1
4280:y
4274:R
4270:y
4263:L
4259:y
4254:y
4237:}
4229:R
4225:y
4218:R
4213:)
4208:y
4205:1
4200:(
4195:)
4192:y
4184:R
4180:y
4176:(
4173:+
4170:1
4164:,
4157:L
4153:y
4146:L
4141:)
4136:y
4133:1
4128:(
4123:)
4120:y
4112:L
4108:y
4104:(
4101:+
4098:1
4089:|
4079:L
4075:y
4068:R
4063:)
4058:y
4055:1
4050:(
4044:)
4040:y
4032:L
4028:y
4023:(
4019:+
4016:1
4010:,
4003:R
3999:y
3992:L
3987:)
3982:y
3979:1
3974:(
3969:)
3966:y
3958:R
3954:y
3950:(
3947:+
3944:1
3938:,
3935:0
3927:{
3923:=
3918:y
3915:1
3887:y
3884:1
3876:x
3873:=
3868:y
3865:x
3844:.
3838:4
3835:/
3832:1
3822:2
3819:/
3816:1
3806:2
3803:/
3800:1
3790:2
3787:/
3784:1
3770:4
3767:/
3764:1
3754:2
3751:/
3748:1
3726:R
3722:Y
3699:R
3695:X
3673:}
3667:R
3663:Y
3652:y
3645:,
3640:R
3636:X
3625:x
3621::
3614:y
3606:x
3595:y
3591:x
3588:+
3585:y
3578:x
3573:{
3562:y
3558:x
3542:R
3538:Y
3532:R
3528:X
3519:R
3515:Y
3511:x
3508:+
3505:y
3500:R
3496:X
3470:}
3464:L
3460:Y
3454:R
3450:X
3443:y
3438:R
3434:X
3430:+
3425:L
3421:Y
3417:x
3414:,
3409:R
3405:Y
3399:L
3395:X
3386:R
3382:Y
3378:x
3375:+
3372:y
3367:L
3363:X
3354:R
3350:Y
3344:R
3340:X
3331:R
3327:Y
3323:x
3320:+
3317:y
3312:R
3308:X
3304:,
3299:L
3295:Y
3289:L
3285:X
3276:L
3272:Y
3268:x
3265:+
3262:y
3257:L
3253:X
3248:{
3244:=
3234:}
3229:R
3225:Y
3216:L
3212:Y
3208:{
3205:}
3200:R
3196:X
3187:L
3183:X
3179:{
3176:=
3169:y
3166:x
3131:.
3127:}
3122:L
3118:Y
3111:x
3108:,
3105:y
3097:R
3093:X
3084:R
3080:Y
3073:x
3070:,
3067:y
3059:L
3055:X
3051:{
3048:=
3045:}
3040:L
3036:Y
3024:R
3020:Y
3013:{
3010:+
3007:}
3002:R
2998:X
2989:L
2985:X
2981:{
2978:=
2975:y
2969:x
2949:,
2947:}
2942:2
2939:/
2936:3
2926:2
2923:/
2920:1
2910:2
2907:/
2904:1
2894:2
2891:/
2888:1
2867:y
2864:=
2861:}
2856:R
2852:Y
2843:L
2839:Y
2835:{
2832:=
2829:}
2824:R
2820:Y
2816:+
2813:0
2805:L
2801:Y
2797:+
2794:0
2791:{
2788:=
2785:y
2782:+
2779:}
2769:{
2766:=
2763:y
2760:+
2757:0
2738:x
2735:=
2732:}
2727:R
2723:X
2714:L
2710:X
2706:{
2703:=
2700:}
2697:0
2694:+
2689:R
2685:X
2678:0
2675:+
2670:L
2666:X
2662:{
2659:=
2656:}
2646:{
2643:+
2640:x
2637:=
2634:0
2631:+
2628:x
2609:0
2606:=
2603:}
2593:{
2590:=
2587:}
2577:{
2574:+
2571:}
2561:{
2558:=
2555:0
2552:+
2549:0
2526:}
2523:Y
2513:y
2509::
2502:y
2498:+
2495:x
2492:{
2489:=
2486:Y
2483:+
2480:x
2476:,
2473:}
2470:X
2460:x
2456::
2453:y
2450:+
2443:x
2439:{
2436:=
2433:y
2430:+
2427:X
2405:,
2402:}
2397:R
2393:Y
2389:+
2386:x
2383:,
2380:y
2377:+
2372:R
2368:X
2359:L
2355:Y
2351:+
2348:x
2345:,
2342:y
2339:+
2334:L
2330:X
2326:{
2323:=
2320:}
2315:R
2311:Y
2302:L
2298:Y
2294:{
2291:+
2288:}
2283:R
2279:X
2270:L
2266:X
2262:{
2259:=
2256:y
2253:+
2250:x
2220:=
2217:}
2207:{
2204:=
2201:}
2191:{
2185:=
2182:0
2169:x
2163:R
2159:X
2152:L
2148:X
2143:x
2123:.
2120:}
2117:S
2111:s
2108::
2105:s
2099:{
2096:=
2093:S
2080:S
2076:S
2062:,
2059:}
2054:L
2050:X
2038:R
2034:X
2027:{
2024:=
2021:}
2016:R
2012:X
2003:L
1999:X
1995:{
1989:=
1986:x
1973:}
1970:R
1966:X
1961:L
1957:X
1953:x
1940:}
1937:R
1933:Y
1928:L
1924:Y
1920:y
1915:}
1912:R
1908:X
1903:L
1899:X
1895:x
1879:i
1875:c
1869:n
1865:S
1860:c
1856:x
1852:R
1848:L
1844:i
1840:c
1836:i
1832:n
1828:x
1824:x
1820:R
1816:L
1812:R
1808:L
1802:i
1798:S
1792:n
1788:i
1783:n
1779:}
1777:R
1773:L
1769:x
1744:2
1741:/
1738:1
1728:2
1725:/
1722:1
1704:-
1702:n
1698:n
1688:2
1685:/
1682:1
1672:2
1669:/
1666:1
1651:x
1647:x
1643:x
1636:x
1627:2
1624:S
1567:α
1563:S
1558:i
1548:i
1544:S
1537:n
1533:S
1524:n
1517:n
1513:S
1506:i
1502:S
1495:n
1491:S
1484:i
1480:S
1473:n
1469:S
1462:i
1458:S
1452:n
1448:i
1423:n
1419:S
1414:n
1406:0
1403:S
1386:i
1382:S
1376:0
1370:i
1353:i
1346:i
1342:S
1335:.
1321:i
1317:S
1311:n
1305:i
1288:n
1284:S
1279:n
1271:.
1263:0
1260:S
1216:.
1214:x
1210:y
1205:x
1200:R
1196:y
1189:R
1185:Y
1180:R
1176:y
1170:.
1168:y
1164:x
1158:L
1154:x
1150:y
1143:L
1139:X
1134:L
1130:x
1120:y
1116:x
1111:}
1108:R
1104:Y
1099:L
1095:Y
1091:y
1086:}
1083:R
1079:X
1074:L
1070:X
1066:x
1031:y
1027:x
1022:x
1018:y
1012:y
1008:x
1002:y
998:x
983:.
980:x
976:y
970:y
966:x
961:y
957:x
924:L
920:R
916:R
912:L
904:}
902:R
898:L
867:R
863:L
859:}
857:R
853:L
847:R
843:L
796:ω
790:ω
788:2
783:ω
771:8
768:/
765:1
755:4
752:/
749:1
739:2
736:/
733:1
720:ω
704:a
698:a
694:R
689:a
683:a
679:L
671:a
667:R
662:a
658:L
652:a
631:4
628:/
625:3
615:2
612:/
609:1
593:4
590:/
587:1
577:2
574:/
571:1
555:2
552:/
549:1
488:R
484:L
473:}
471:R
467:L
454:4
451:/
448:2
438:2
435:/
432:1
416:R
406:L
401:}
389:}
387:R
383:L
374:R
370:L
366:}
364:R
360:L
354:R
350:L
346:R
342:L
338:b
334:a
329:a
325:b
319:b
315:a
310:b
306:a
292:2
289:L
285:1
282:L
274:2
271:L
267:1
264:L
262:(
257:}
255:R
251:L
245:)
243:R
239:L
237:(
233:R
229:L
196:α
191:α
183:α
179:η
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
