# Surreal Numbers

The wikipedia entry for surreal numbers is a nice clear description of what you get out of Conway's definitions. The present page is an old attempt at describing the surreals for myself; it is superseded by a newer version and shall be retired in due course.

It is possible to describe the surreals as mappings from ordinals to a two-member set, to be thought of as {+,-}. To interpret a mapping ({+,-}: f |n) with n ordinal as a surreal, first identify the maximal ordinal m subsumed by n for which ({+}:f|m) or ({-}:f|m), i.e. the first m entries in f are identical. We then interpret (:f:m) as m if ({+}:f|m) or as -m if ({-}:f|m); if m is 0 these both interpret (:f:m) as 0. If n > m, we know f(m) is the member of {+,-} that isn't in (|f:m); and interpret f as (:f:m) + sum(:f(i+m)/power(i,2) ←i:). But I prefer to start from Conway's formulation, albeit transposed from his (which is optimised for discussion of games) to mine (which, for all its deficiencies, is at least consistent across everything I write).

Logically inverting various conditions in Conway's strongly inductive definition and converting to my notations, I can define a surrogate for the ordering on the surreal numbers by: < relates a to b precisely if

• a and b are atoms: write a = W ← X and b = Y ← Z,
• W, X, Y and Z are collections,
• < subsumes (W:all:X) and (Y:all:Z),
• there is either some y in Y for which not(y<a) or some x in X for which not(b<x); i.e. not( (Y|<{a}) and ({b}:<|X) ).

The first three conditions just tell us what <'s values are. An atom is simply a relation with exactly one left value and exactly one right value; in effect, it is used here simply to implement a pair. A collection is a relation for which, if it relates x to y, then x and y are equal. Note that we can't simplify the above by replacing a = W ← X with (W:all:X) – and likewise for b – because (W:all:X) is empty if either W or X is empty, regardless of the other; the above distinguishes far more atoms with an empty value than the (W:all:X) form would.

When either W or X is empty, hence so is (W:all:X), and every relation subsumes empty, hence < subsumes (W:all:X) trivially. If we now define

• 0 = {}←{}
• 1 = {0}←{}
• -1 = {}←{0}

we see that any pair of them satisfies the first three conditions for < and it remains to consider the final one. This requires a non-empty value for at least one of X and Y: we can thus infer that < does not relate 0 to 0, 1 to 0, 0 to -1 or 1 to -1 (since, in each case, X and Y would be empty). With a = -1 we have 0 in X so can apply not(0<0) to infer -1 < 0 and apply not(1<0) to infer -1 < 1; when b is also -1, Y is empty and X's only member is x = 0, for which b<x so it doesn't serve to satisfy the final condition, so we infer not(-1 < -1). With b = 1, we have 0 in Y so can likewise infer 0 < 1 and -1 < 1; when a is also 1, X is empty and Y's only member is y = 0, for which y<a so we infer not(1<1). Thus the comparisons among 0, 1 and -1 all come out the way their names would suggest; and we have these three values (at least) as values of <.

From this we can construct an equivalence S defined by: S relates a to b precisely if

• a and b are (left or right) values of < but
• neither a<b nor b<a.

The equivalence classes of S may then be construed as the surreal numbers. It remains to prove that < is transitive and has no fixed points; then to prove that S is an equivalence.

## Arithmetic

We can define an addition on <'s values by, with x = W←X, y = Y←Z:

• x+y = {a+y: a in W}&unite;{x+b: b in Y} ← {a+y: a in X}&unite;{x+b: b in Z}

It is then necessary to show that < subsumes x+y and that replacing either x or y with some S-equivalent value of < yields a value S-equivalent to x+y, to make the above an addition on surreal numbers. We must likewise do similar for the addition defined by (for the same x and y as above):

• x.y = {a.y +x.b -a.b: either a in W and b in Y or a in X and b in Z} ← {a.y +x.b -a.b: either a in W and b in Z or a in X and b in Y}

Once it is established that the arithmetic is well-defined, we should prove that it has the familiar properties of an ordered field:

• addition must form a group; 0 should be it's identity
• when 0 is left out, multiplication must form a group
• x + y = y + x
• x.y = y.x
• x<y implies x+a < y+a
• x< y, 0<a implies x.a < y.a
• x< y, a<0 implies y.a < x.a
• x + (y + z) = (x + y) + z
• x.(y.z) = (x.y).z
• x.(y+z) = (x.y) + (y.z)

and, indeed, we'll need to prove some of these (the ones not involving multiplication) in order to establish that our multiplication is well-defined.

For multiplication, begin by noting that 0.(Y←Z) and (W←X).0 each quantify over an empty collection, in each term of the x.y specification, so yield {}←{} = 0. Thus 0.y = 0 = x.0 for all surreal x, y. Next, notice that (-1).(-1) = ({}←{0}).({}←{0}) has W = {} = Y and X = {0} = Z; so that only the a in X and b in Z clause of the definition can be satisfied, and it has a = 0 = b yielding a.y +x.b -a.b = 0.y +x.0 -0.0 = 0+0-0 = 0. Thus (-1).(-1) = {0}←{} = 1. Likewise, 1.1's only non-empty term is a in W and b in Y with a = 0 = b yielding 1.1 = {0}←{} = 1. Thus -1 and 1 each squares to 1.

## Measure

In June 2005 I had Steve Thornton gabbling incoherently at me about the surreals, which he did not appear to understand; in my attempts to make sense of what he was trying to say, which seemed to involve a changed reading of the left and right sets of a number, I got to wondering about using measures in place of collections.

We can transparently represent any finite collection of surreals as a discrete measure on the surreals which integrates any function on the surreals to yield the sum of its values at the collection's members. We could then build number-like things each of which is a pair of measures. The requirement, for W←X to be a value of <, that < subsume (W:all:X) can be expressed as all w in W, x in X satisfy w<x; which we must turn into a constraint on our pair of measures. Likewise, we need to transform not( (Y|<:{a}) and ({b}:<|X) ) into some measure form. Now that I've refreshed my memory of the standard surreals, I can't immediately think of a nice way to do that.

Written by Eddy.
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