Conway's real closed field No of surreal numbers is a sweeping generalization of the real numbers and the ordinals to which a number of elementary functions such as log and exponentiation have been shown to extend. The problems of identifying significant classes of functions that can be so extended and of defining integration for them have proven to be formidable. In this paper we address this and related unresolved issues by showing that extensions to No, and thereby integrals, exist for most functions arising in practical applications. In particular, we show they exist for a large subclass of the resurgent functions, a subclass that contains the functions that at ∞ are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as solutions to nonresonant linear and nonlinear meromorphic systems of ODEs or of difference equations. By suitable changes of variables we deal with arbitrarily located singular points. We further establish a sufficient condition for the theory to carry over to ordered exponential subfields of No more generally and illustrate the result with structures familiar from the surreal literature. The extensions of functions and integrals that concern us are constructive in nature, which permits us to work in NBG less the Axiom of Choice (for both sets and proper classes). Following the completion of the positive portion of the paper, it is shown that the existence of such constructive extensions and integrals of substantially more general types of functions (e.g. smooth functions) is obstructed by considerations from the foundations of mathematics.

AbstractSince Euclid defined a point as “that which has no part” it has been widely assumed that points are necessarily unextended. It has also been assumed that this is equivalent to saying that points or, more properly speaking, degenerate segments, have length zero. We challenge these assumptions by providing models of Euclidean geometry where the points are extended despite the fact that the degenerate segments have null lengths, and observe that whereas the extended natures of the points are not recognizable in the given models, they can be recognized and characterized by structures that are suitable expansions of the models.

John Conway challenged and inspired us in many different ways. His influence can be seen across so many areas of mathematics that a complete account would fill many volumes, which would continue to expand forever. These three articles present very different parts of mathematics where modern approaches are built on foundations laid by John.

AbstractIn 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field${\mathbf {No}}$of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered$K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of${\mathbf {No}}$, i.e. a subfield ($K$-subspace) of${\mathbf {No}}$that is an initial subtree of${\mathbf {No}}$. In this sequel, analogous results are established forordered exponential fields, making use of a slight generalization of Schmeling’s conception of atransseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of$({\mathbf {No}}, \exp )$. These include all models of$T({\mathbb R}_W, e^x)$, where${\mathbb R}_W$is the reals expanded by aconvergent Weierstrass system W. Of these, those we calltrigonometric-exponential fieldsare given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of${\mathbf {No}}$, which includes${\mathbf {No}}$itself, extend tocanonicalexponential functions on theirsurcomplexcounterparts. The image of the canonical map of the ordered exponential field${\mathbb T}^{LE}$oflogarithmic-exponentialtransseries into${\mathbf {No}}$is shown to be initial, as are the ordered exponential fields${\mathbb R}((\omega ))^{EL}$and${\mathbb R}\langle \langle \omega \rangle \rangle $.

The purpose of this chapter is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua. Among the theories we will consider are those that emerge from nonstandard analysis, nilpotent infinitesimalist approaches to portions of differential geometry and the theory of surreal numbers. Since these theories have roots in the algebraic, geometric and analytic infinitesimalist theories of the late nineteenth and early twentieth centuries, we will also provide overviews of the latter theories and some of their relations to the contemporary ones. We will find that the contemporary theories, while offering novel and possible alternative visions of continua, need not be (and in many cases are not) regarded as replacements for the Cantor-Dedekind theory and its corresponding theories of analysis and differential geometry.

We consider derivations $\partial$ on Conway's field $\mathbf{No}$ of surreal numbers such that the ordered differential field $(\mathbf{No},\partial)$ has constant field $\mathbb{R}$ and is a model of the model companion of the theory of $H$-fields with small derivation. We show that this determines $(\mathbf{No},\partial)$ uniquely up to isomorphism, and that this structure is absolutely homogeneous universal for models of this theory with constant field $\mathbb{R}$.

The purpose of this paper is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua. Among the theories we will consider are those that emerge from nonstandard analysis, nilpotent infinitesimalist approaches to portions of differential geometry and the theory of surreal numbers. Since these theories have roots in the algebraic, geometric and analytic infinitesimalist theories of the late nineteenth and early twentieth centuries, we will also provide overviews of the latter theories and some of their relations to the contemporary ones.

Introduction . In his monograph On Numbers and Games [7], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including ω, ω , /2, 1/ ω , and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No , is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered “number” fields—be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with Global Choice , henceforth NBG [cf. 21, Ch. 4], it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields. This can be made precise by saying that whereas the ordered field of reals is (up to isomorphism) the unique homogeneous universal Archimedean ordered field, No is (up to isomorphism) the unique homogeneous universal orderedfield [14]; also see [10], [12], [13]. However, in addition to its distinguished structure as an ordered field, No has a rich hierarchical structure that (implicitly) emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or simplicity hierarchy , as we have called it [15], depends upon No's (implicit) structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with No's structure as an ordered group and an ordered field, respectively, it being understood that x is simpler than y just in case x is a predecessor of y in the tree.

New striking analogies between H. Hahn’s fields of generalised series with real coefficients, G. H. Hardy’s field of germs of real valued functions, and J. H. Conway’s field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery.

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field No of surreal numbers containing the reals and the ordinals, as well as a vast array of less familiar numbers. A longstanding aim has been to develop analysis on No as a powerful extension of ordinary analysis on the reals. This entails finding a natural way of extending important functions f from the reals to the reals to functions f* from the surreals to the surreals, and naturally defining integration on the f*. The usual square root, log, and exp were naturally extended to No by Bach, Conway, Kruskal, and Norton, retaining their usual properties. Later Norton also proposed a treatment of integration, but Kruskal discovered it has flaws. In his recent survey [2, p. 438], Siegel characterizes the question of the existence of a reasonable definition of surreal integration as "perhaps the most important open problem in the theory of surreal numbers." This paper addresses this and related unresolved issues with positive and negative results. In the positive direction, we show that semi-algebraic, semi-analytic, analytic, meromorphic, or more generally Écalle-Borel transseriable functions extend naturally to No, and an integral with good properties exists on them. In the negative direction, we show there is a fundamental set-theoretic obstruction to naturally extending many larger families of functions.

In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field No of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of No, i.e. a subfield of No that is an initial subtree of No. In this sequel to [15], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of No that are themselves initial. It is further shown that an initial subdomain of No is discrete if and only if it is an initial subdomain of No's canonical integer part Oz of omnifc integers. Finally, extending results of [15], the theories of divisible ordered abelian groups and real-closed ordered fields are shown to be the sole theories of ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of No.

We suggest that, far from establishing an inconsistency in the standard theory of the geometrical linear continuum, Zeno’s Paradox of Extension merely establishes an inconsistency between the standard theory of geometrical magnitude and a misguided system of length measurement. We further suggest that our resolution of Zeno’s paradox is superior to Adolf Grünbaum’s now standard resolution based on Lebesgue measure theory.

AbstractIn his monographOn Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including −ω, ω/2, 1/ω,and ω − π to name only a few. Indeed, this particular real-closed field, which Conway callsNo, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG (von Neumann–Bernays–Gödel set theory with global choice), it may be said to contain “All Numbers Great and Small.” In this respect,Nobears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields.In Part I of the present paper, we suggest that whereas ℝ should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom),Nomay be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying frameworkNoprovides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis.In addition to its inclusive structure as an ordered field, the systemNoof surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development ofNooutlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features ofNoare brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations ofNoas an absolute continuum.

Each surreal number has a unique Conway name (or normal form) that is characteristic of its individual properties.The paper answers the following two questions that are naturally suggested by the surreal number system's structure as a lexicographically ordered full binary tree.(i) Given the Conway name of a surreal number, what are the Conway names of its two immediate successors?(ii) Given the Conway names of the members of a chain of surreal numbers of limit length, what is the Conway name of the immediate successor of the chain?

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We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an E-number. In that case, this field is even closed under surreal exponentiation, and is an elementary extension of the real exponential field.