×

Foundations of analysis over surreal number fields. (English) Zbl 0621.12001

North-Holland Mathematics Studies, 141. Notas de Matemática, 117. Amsterdam etc.: North-Holland. XVI, 373 p.; $ 66.75; Dfl. 150.00 (1987).
In his famous book ”On numbers and games” (1976; Zbl 0334.00004), J. H. Conway constructed a field \({\mathfrak No}\) by an inductive process. \({\mathfrak No}\) is a proper class and the induction is over the class \({\mathfrak On}\) of all ordinal numbers. \({\mathfrak No}\) is the field of surreal numbers. Every surreal number x is defined at some step \(\alpha\in {\mathfrak On}\), called the birthday of x, denoted by b(x). For every ordinal number \(0<\xi\) such that the cardinal number \(\omega_{\xi}\) is regular, \(\xi\) \({\mathfrak No}\) is defined to be the set of \(x\in {\mathfrak No}\) with \(b(x)<\omega_{\xi}\). \({\mathfrak No}\) and \(\xi\) \({\mathfrak No}\) are real closed field. \(\xi\) \({\mathfrak No}\) is even an \(\eta\) \({}_{\xi}\)-field [cf. the author, Trans. Am. Math. Soc. 103, 341-352 (1962; Zbl 0108.257)]. The fields \(\xi\) \({\mathfrak No}\) are also called surreal number fields. It is important to note that (apart from the structure present in every real closed field) the fields of surreal numbers carry additional structure provided by the birthday function. A striking example is
Conway’s Simplicity Theorem. Every nonempty interval in a surreal number field contains a unique number born before all other elements of the interval (p. 124).
The objective of this book is to lay foundations for doing analysis over surreal number fields. The main results are concerned with convergence of formal power series. The mode of convergence considered is called hyper- convergence: If K is a field and G is an ordered abelian group then K((G)) is the field of formal power series with exponents in G and coefficients in K. For every formal power series \(A(X)=\sum_{n\geq 0}a_ nX^ n\in K[[ X]]\) and every x in the valuation ideal of K((G)), it makes sense to define \(A(x)=\sum_{n\geq 0}a_ nx^ n\in K((G))\). A(x) is considered as the limit at x of the formal series A(X) with respect to hyper-convergence. This is applicable to surreal number fields since, because of the birthday function, every surreal number field is a power series field in a natural way.
The book starts with a very broad exposition of some basic properties of ordered sets, ordered groups, ordered fields. To be able to do meaningful topology over the field \(\xi\) \({\mathfrak No}\) the notion of a \(\xi\)-topology is introduced. This is a restricted topology much in the same spirit as the semi-algebraic topology of a semi-algebraic set used by H. Delfs and M. Knebusch in their development of semi-algebraic geometry [Math. Z. 178, 175-213 (1981; Zbl 0447.14003)]. Some very basic properties of the \(\xi\)-topology are discussed. The surreal number fields are defined and their most important properties are discussed. A very general introduction to valuation theory and to power series fields is given. This is applied to surreal number fields. Hyper-convergence is studied for a rather general class of power series fields containing the class of surreal number fields. There are no results which are really peculiar to surreal number fields. Finally, based on the notion of hyper- convergence, there is a short discussion of analytic functions of a surreal variable.
Readers of the book should always be aware that there are very numerous misprints.
Reviewer: N.Schwartz

MSC:

12-02 Research exposition (monographs, survey articles) pertaining to field theory
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
12J10 Valued fields
26E99 Miscellaneous topics in real functions
14Pxx Real algebraic and real-analytic geometry
12J15 Ordered fields