Surreal numbers

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 $0$ 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].

References

[a1]	N.L. Alling, "Foundations of analysis over surreal number fields" , North-Holland (1987)
[a2]	J.H. Conway, "On numbers and games" , Acad. Press (1976)
[a3]	N. Cuesta Dutari, "Algebra ordinal" Rev. Acad. Ciencis. Madrid , 48 (1954) pp. 103–145
[a4]	H. Gonshor, "An introduction to the theory of surreal numbers" , Cambridge Univ. Press (1986)
[a5]	H. Hahn, "Über die nichtarchimedischen Grössensysteme" S. Ber. Akad. Wiss. Wien, Math. Naturw. Abt. IIa , 116 (1907) pp. 601–655
[a6]	J.E. Kelley, "General topology" , v. Nostrand (1955)
[a7]	D.E. Knuth, "Surreal numbers" , Addison-Wesley (1974)