I think you just have to look at the collection of Cauchy sequences where “sequence” means a function from the ordinals to the surreals, and “Cauchy” means that the terms eventually get smaller than any surreal.
I’d be skeptical of that assertion. Even sticking to ordinary topology on actual sets, transfinite sequences are not enough to do limits in general; in general you need nets. (Or filters.) Doesn’t mean you’ll need that here—might the fact that the surreals are linearly ordered help? -- but I don’t think it’s something you should assume would work.
But yeah it does seem like you’ll need something able to contain a “sequence” of order type that of the class of all ordinals; quantifying over ordinals or surreals or something in the “domain”. (Like, as I said above, limits of surreal-valued functions of a surreal variable shouldn’t pose a problem.)
In any case, sequences or nets are not necessarily the issue. This still doesn’t help with infinite sums, because those are still just ordinary omega-sequences. But really the issue is integration; infinite sums can be ignored if you can get integration. Does the “domain” there have sufficient granularity? Well, uh, I don’t know.
I think you just have to look at the collection of Cauchy sequences where “sequence” means a function from the ordinals to the surreals, and “Cauchy” means that the terms eventually get smaller than any surreal.
I’d be skeptical of that assertion. Even sticking to ordinary topology on actual sets, transfinite sequences are not enough to do limits in general; in general you need nets. (Or filters.) Doesn’t mean you’ll need that here—might the fact that the surreals are linearly ordered help? -- but I don’t think it’s something you should assume would work.
But yeah it does seem like you’ll need something able to contain a “sequence” of order type that of the class of all ordinals; quantifying over ordinals or surreals or something in the “domain”. (Like, as I said above, limits of surreal-valued functions of a surreal variable shouldn’t pose a problem.)
In any case, sequences or nets are not necessarily the issue. This still doesn’t help with infinite sums, because those are still just ordinary omega-sequences. But really the issue is integration; infinite sums can be ignored if you can get integration. Does the “domain” there have sufficient granularity? Well, uh, I don’t know.