Surreal number
Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers.
Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.
The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.
[1] The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations.
Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers.
[2] Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.
A separate route to defining the surreals began in 1907, when Hans Hahn introduced Hahn series as a generalization of formal power series, and Felix Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure.
In 1962, Norman Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α and, in 1987, he showed that taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.
[4] If the surreals are considered as 'just' a proper-class-sized real closed field, Alling's 1962 paper handles the case of strongly inaccessible cardinals which can naturally be considered as proper classes by cutting off the cumulative hierarchy of the universe one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense.
After an infinite number of stages, infinite subsets become available, so that any real number a can be represented by { La | Ra }, where La is the set of all dyadic rationals less than a and Ra is the set of all dyadic rationals greater than a (reminiscent of a Dedekind cut).
(The set union expression appears in our construction rule, rather than the simpler form Sn−1, so that the definition also makes sense when n is a limit ordinal.)
The equivalence classes at each stage n of induction may be characterized by their n-complete forms (each containing as many elements as possible of previous generations in its left and right sets).
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 shown that the definitions of negation, addition and multiplication are consistent, in the sense that: With these rules one can now verify that the numbers found in the first few generations were properly labeled.
No individual Sn is closed under addition and multiplication (except S0), but S∗ is; it is the subring of the rationals consisting of all dyadic fractions.
Define Sω as the set of all surreal numbers generated by the construction rule from subsets of S∗.
One can determine the relationship between ω and ε by multiplying particular forms of them to obtain: This expression is well-defined only in a set theory which permits transfinite induction up to Sω2.
The rationals are not an identifiable stage in the surreal construction; they are merely the subset Q of Sω containing all elements x such that x b = a for some a and some nonzero b, both drawn from S∗.
[9] An equivalent of Cauchy sequences can be defined as well, although they have to be indexed by the class of ordinals; these will always converge, but the limit may be either a number or a gap that can be expressed as
Its values are completely determined by the basic relation xy+z = xy · xz, and where defined it necessarily agrees with any other exponentiation that can exist.
For x negative infinite the odd-numbered partial sums are strictly decreasing and the [x]2n+1 notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.
This is well-defined for all surreal arguments (the value exists and does not depend on the choice of zL and zR).
Using this definition, the following hold:[e] The surreal exponential is essentially given by its behaviour on positive powers of ω, i.e., the function
For example, in Go, the board will slowly fill up with pieces until there are just a few small islands of empty space where a player can move.
He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their disjunctive sum.
Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.
Alternative approaches to the surreal numbers complement Conway's exposition in terms of games.
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).
A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich.
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., ν(ω) = −1.
