# Integral of a function defined in the set of Surreal Numbers

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

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The surreal numbers form a proper class, so $C$ is not a set to begin with, therefore $\{C\}$ is not an object you can write. –  Asaf Karagila Feb 23 '12 at 16:29

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.

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There is an interesting paper on that by Antongiulio Fornasier, thesis is available online. That's a big work but I believe the subject is worth the time.

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The recent paper, "An Analogue of Real Analysis for Surreal Numbers" has information on integration. http://arxiv.org/pdf/1307.7392v1.pdf

[math.CA] 28 Jul 2013

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Here is a PhD thesis which answers your questions. It's pretty dense though! The answer is yes, and it turns out that log integrates as one would expect but that exp is a bit more tricky.

http://www.dm.unipi.it/~fornasiero/phd_thesis/thesis_fornasiero_linearized.pdf

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