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## An extract on #sonu

In what is now called the sign-expansion or sign-sequence of a surreal number, a surreal number is a function whose domain is an ordinal and whose codomain is { 1, +1 }. This is equivalent to Conway's L-R sequences. Define the binary predicate "simpler than" on numbers by x is simpler than y if x is a proper subset of y, i.e. if dom(x) < dom(y) and x() = y() for all < dom(x). 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 x < y if one of the following holds: x is simpler than y and y(dom(x)) = + 1; y is simpler than x and x(dom(y)) = 1; there exists a number z such that z is simpler than x, z is simpler than y, x(dom(z)) = 1 and y(dom(z)) = + 1. Equivalently, let (x,y) = min({ dom(x), dom(y)} { : < dom(x) < dom(y) x() y() }), so that x = y if and only if (x,y) = dom(x) = dom(y). Then, for numbers x and y, x < y if and only if one of the following holds: (x,y) = dom(x) (x,y) < dom(y) y((x,y)) = + 1; (x,y) < dom(x) (x,y) = dom(y) x((x,y)) = 1; (x,y) < dom(x) (x,y) < dom(y) x((x,y)) = 1 y((x,y)) = + 1. For numbers x and y, x y if and only if x < y x = y, and x > y if and only if y < x. Also x y if and only if y x. The relation < is transitive, and for all numbers x and y, exactly one of x < y, x = y, x > y, holds (law of trichotomy). This means that < is a linear order (except that < is a proper class). For sets of numbers, L and R such that x L y R (x < y), there exists a unique number z such that x L (x < z) y R (z < y), For any number w such that x L (x < w) y R (w < y), w = z or z is simpler than w. Furthermore, z is constructible from L and R by transfinite induction. z is the simplest number between L and R. Let the unique number z be denoted by (L,R). For a number x, define its left set L(x) and right set R(x) by L(x) = { x| : < dom(x) x() = + 1 }; R(x) = { x| : < dom(x) x() = 1 }, then (L(x),R(x)) = x. 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 as particular cases of surreals. However, similar definitions can be made that eliminate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule g dom f (h dom g (h dom f )) and whose range is { , + }. "Simpler than" is very simply defined nowx is simpler than y if x dom y. The total ordering is defined by considering x and y as sets of ordered pairs (as a function is normally defined): Either x = y, or else the surreal number z = x y is in the domain of x or the domain of y (or both, but in this case the signs must disagree). We then have x < y if x(z) = or y(z) = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom f in order of simplicity (i.e., inclusion), and then write down the signs that f assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is { + }.