Surreal numbers are a creation of the British mathematician
J.H. Conway
[a2].
They find their origin in the area of game
theory. Their description can be found in Conway's book
[a2]
(1976),
but two years earlier
D.E. Knuth
had already
popularized the surreal numbers in his mathematical novelette
[a7].
Only in later years have surreal numbers become the
subject of more traditional mathematical papers and books
[a4],
[a1].
Surreal numbers are obtained by turning the Dedekind construction for
the real numbers out of the rational numbers (cf.
Dedekind cut)
into a powerful mechanism which can be used to create all numbers from
nothing. Starting with the empty set, one introduces in stages new numbers defined
in terms of partitions of the set of existing numbers in two
ordered parts. In this way new surreal numbers are created, all sharing the same
birthday.
The construction is repeated for each ordinal birthday,
including the transfinite ones. In fact, the construction
never terminates, and therefore the surreal numbers do not
form a set but a proper class (cf.
Types, theory of).
In the finite stages the construction will yield objects which play the
role of the finite dyadic fractions in the rational numbers.
The remaining finite rationals will appear together with the other real numbers on day
,
but on this day also the first transfinite numbers
and the first infinitesimals will be created. Subsequently
more and more standard numbers will be added.
What makes the construction interesting is the fact that it is possible
to provide, together with the inductive definition of the objects,
also inductive definitions of order, equality and algebraic operations
like addition and multiplication. Equipped with these operations the
domain of surreal numbers behaves like a real-closed field (except for the
fact that it is not a set but a proper class).
In fact, the definition of the order and that of the objects are
tightly connected. A number is a pair of sets of numbers with the
property that no member of its left set is larger or equal to
a member of its right set. A number is less or equal to another number in
case no member of the left set of the first number is less or equal
to the second number and no member of the right set
of the second number is less or equal to the first.
The above definitions look rather circular, and consequently it is
hard to grasp the intuition behind these definitions, but Knuth's novelette
shows that the definitions make sense and that their meaning can
be explained to a general audience as well. Still, both Knuth and
Conway leave the task of filling out the details of the
proof and the construction to the reader for purposes of enjoyment; only
the subsequent authors felt compelled to work out and publish all the hard mathematics.
It turns out that the objects defined form a
pre-order:
different objects can be both smaller or equal and larger or equal, and
consequently the ordered structure is obtained only
after factoring out the corresponding equivalence relation.
The definitions of addition and multiplication follow the same inductive pattern. For
example, the left set of the sum consists of all sums of
one number and a left-part number of the other, whereas the right part of
the sum consists of the sums of one number and a right part of the other.
Proving something about the structure amounts in almost-all cases to total induction (cf.
Induction axiom)
on everything, or, in some cases, on
transfinite induction
on the day of birth. These inductions are based
on the well-foundedness of the sets constructed (cf. also
Well-founded relation)
or on the fact that on day
nothing has yet been created.
Notwithstanding the fact that the initial publications on surreal numbers
originate from game theory (Conway) and recreational mathematics (Knuth),
the surreal numbers relate to serious mathematics.
N.L. Alling
has written a book
[a1]
in which he relates the surreal numbers to classic results of
H. Hahn
[a5]
on ordered vector spaces; such spaces are subfields
of lexicographically ordered sums of the rationals or reals over an index set which itself
is an ordered Abelian group. This corresponds to some alternative representations of the surreal numbers
which can be given in terms of formal power series over
an index
"set"
which is isomorphic to the surreal numbers themselves.
The Dedekind construction was invented to close the gaps in between the rational numbers. For
the surreal numbers the matter is more complicated. The cuts used for their construction, called
Cuesta Dutari cuts
by Alling (after
[a3]),
introduce new gaps while filling previous ones. At no specific stage
will the result be a connected topological domain. However, at the
price of weakening the axioms of set-theoretical topology (by requiring the open
sets to be closed under union up to a bounded cardinality only)
there will exist birthdays for which the domain as constructed up to
that day will behave like a connected set in this weaker topology.
The classical theory of analysis of power series can be generalized
to a large extent for the surreal numbers. In fact, convergence properties are sometimes better
than for the traditional real field. Conway has observed,
for example, that each formal multivariate power series with
real coefficients will be absolutely convergent for all infinitesimal values of the
variables. The notion of differentiation and that of a derived function
generalizes as well; these notions behave like in elementary calculus.
A pervading complication in Alling's book remains the fact that the surreal
numbers do not form a set but a proper class. For
Conway this is one more argument in support of his
Mathematicians' Liberation Movement:
free constructions should be permitted as long as it remains
good mathematics and mathematicians should never tie their hands by
restricting themselves to any specific foundation of mathematics. Alling has shown
that such a call to revolution for the case of
the surreal numbers is not needed; his treatment is
fully contained within the framework of Kelley–Morse set theory
[a6].