FOM: Surreal Numbers

[John Pais]

Joe Shipman wrote:

> Simpson:
> >I glanced through ``On Numbers and
> >Games'' some more, but I didn't find where Conway states or proves a
> >completeness property of the surreal numbers.  Could someone please
> >help me with this?
>
> This "property" is actually what defines the surreal numbers.  The
> "Construction" on page 4, which comes before any theorems, is simply
>
> If L,R are any two sets of numbers, and no member of L is >= any member
> of R, then there is a number { L | R }.  All numbers are constructed in
> this way.
>

Harry Gonshor's book "Introduction to the Theory of Surreal Numbers," CUP
1986, provides an interesting development of surreal numbers based on the
usual set theoretic framework.

p. 1:

"I was introduced to this subject in a talk by M.D. Kruskal ... in 1977.
Since then I have developed the subject from a somewhat different foundation
from Conway, and carried it further in several directions. I define the
surreal numbers as objects which are rather concrete to most mathematicians,
as compared to Conway's, which are equivalence classes of inductively
defined objects.

The surreal numbers form a proper class which contains the real numbers and
the ordinals among other things. ... Instead of being compelled to create
new entities at each stage and make new definitions, we have unified
definitions at the beginning and obtain the reals as a subsystem of what we
already have. ... more important than obtaining a new way of building up a
familiar set such as the real numbers, is the enrichment of mathematics by
the inclusion of a new structure with interesting properties."

p. 3:

Definition. A *surreal number* is a function from an initial segment of the
ordinals into the set {+,-}, i.e. informally, an ordinal sequence consisting
of pluses and minuses which terminate.

The length, len(a), of a surreal number a is the least ordinal alpha for
which it is undefined.

For stylistic reasons I shall occasionally say that a(alpha)=0 if a is
undefined at alpha.

Definition. If a and b are surreal numbers we define an order as follows: a
< b if a(alpha) < b(alpha) where alpha is the first place where a and b
differ, with the convention that - < 0 < +, e.g. (+-) < (+) < (++).

p. 5:

Theorem 2.1. Let F and G be two sets of surreal numbers such that a in F and
b in G imply a < b. Then there exists a unique c of minimal length such that
both a in F implies a < c and b in G implies c < b. Furthermore, c is an
initial segment of any surreal number strictly between F and G.

Gonshor comments that it is this theorem that makes his alternative
approach  work, and notes that since for Conway's definition different pairs
can give rise to the same number, Conway needs a more "abstract" version
including an inductively defined equivalence relation.

p. 7:

Definition. F|G is the unique c of minimal length such that F < c < G.

p. 9:

Definition. (F',G') is cofinal in (F,G) if for all a in F there exists b in
F' with b >= a and for all a in G there exists b in G' with b <= a.

p. 10:

Theorem 2.6. (the cofinality theorem) Suppose F|G = a, F'< a < G', and
(F',G') is confinal in (F,G), then F'|G' = a.


Theorem 2.8. Let a be a surreal number. Suppose that F'={b: b < a and b is
an initial segment of a} and G'={b: b > a and b is initial segment of
a}.Then a = F'|G'. (In the sequel F'|G' will be called the canonical
representation of a.)


Hopefully, the above provides enough detail to get a feeling for the main
tools Gonshor needs to develop the surreals from first principles set
theoretically. So, now let's jump to how the reals fall out.

p. 33:

Definition. A real number is a surreal number a which is either of finite
length, or is of length omega and satisfies: for all n0 there exists n1, n2
with n1 >= n0, n2 >= n0, a(n1) = +, and a(n2) = -.

So, real numbers aren't constant on an infinite tail, and now instead of
using a Dedekind cut construction, Gonshor can just prove a theorem and
inherit the properties he wants from the surreals.

Lemma 4.3. Let F and G be non-empty sets of dyadic fractions [surreal
numbers of finite length] such that F < G, F has no maximum, and G has no
minimum. Then F|G is a real number.

p. 39:

Theorem 4.3. The real numbers form an ordered field with the l.u.b. property
(i.e. they are essentially the same as the "reals" defined in more
traditional ways).


Next, Gonshor develops a normal form for surreal numbers and uses this to
show that the surreal numbers form a real closed field. Further, he obtains
results on the upper bounds of the lengths of surreal numbers which permits
him to specialize some of his results to sets.

p. 95:

"... This will allow us to obtain subclasses of the proper class of surreal
numbers which are actually *subsets* and closed under
the desirable operations."


Here is one of his main results along these lines, which addresses some
questions and conjectures Steve has made, and which Gonshor obtains
ultimately using "a kind of back and forth argument."

p. 97:

Theorem 6.4. If the cardinality of the lengths of all the coefficients in a
polynomial of odd degree is bounded above by an infinite d, then the
polynomial has a root b such that |len(b)| <= d.

p. 103:

"... Finally, as a culmination of the results of this chapter we have shown
that the subset of surreal numbers a such that |len(a)| <= d for any fixed
infinite cardinal d is a real closed field. ... These are all "honest"
fields since their carriers are *sets*.

Not all subfields have the above form. In fact, the two most well-known
fields found in nature, the rationals and reals, both consist of all
surreals of finite length together with some but not all surreals of length
omega.

The field of all surreals of countable length should be a worthwhile object
for further study."

Gonshor has other interesting and deep results including the development of
a "surreal" theory of generalized epsilon numbers. However, I must end this
already too long posting.

John Pais