Integral of a function defined in the set of Surreal Numbers

Question

Given ${\{C}\}\ $ the set of all the $Surreal\ numbers$, is it possible to define the integral: $$\int_a^b{dxf(x)}$$where $$a\in{\{C}\},b\in{\{C}\},x\in{\{C}\}$$ Thanks

Answer 1

There is some discussion of this topic at the very end of the second edition (2000) On Numbers and Games by Conway. He describes work by himself, Simon Norton, and Martin Kruskal to define integration. According to the description, it looked good for a while, producing workable logarithm function in terms of the integral of $x^{-1}$, but then got stuck, and finally:

For twenty years we believed that the definition was nevertheless probably “correct” in some natural sense, and that these difficulties arose merely because we did not understand exactly which genetic definitions were “legal” to use in it.

Kruskal has now made some progress of a rather sad kind by showing that this belief was false. Namely, the definition integrates $e^t$ over the range $[0, \omega]$ to the wrong answer, $e^\omega$, rather than $e^\omega-1$, independent of whatever reasonable genetic definition we give for the exponential function.

(Page 228.)

There is other discussion of the details in the same section. I do not know whether any progress has been made since then.

Answer 2

As of yesterday, there may be a new development.

“Integration on the Surreals” (Ovidiu Costin, Philip Ehrlich):

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 infinity are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as generic solutions to linear and nonlinear systems of ODEs possibly having irregular singularities. 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. We work in NBG less the Axiom of Choice (for both sets and proper classes), with the result that the extensions of functions and integrals that concern us here have a "constructive" nature in this sense. In the Appendix 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.

The paper has not been published or formally peer-reviewed. But even if it doesn't hold up, its bibliography will be valuable to someone interested in this topic.