So, I haven’t really read this in any detail, but—I am very, very wary of the use of hyperreal and/or surreal numbers here. While as I said I haven’t taken a thorough look at this, to me these look like “well we need infinitesimals and this is what I’ve heard of” rather than having any real reason to pick one of these two. I seriously doubt that either is a good choice.
Hyperreals require picking a free ultrafilter; they’re not even uniquely defined. Surreal numbers (pretty much) completely break limits. (Hyperreals kind of break limits too, due to being of uncountable cofinality, but not nearly as extensively as surreal numbers do, which are of proper-class cofinality.) If you’re picking a number system, you need to consider what you’re actually going to do with it. If you’re going to do any sort of limits or integration with it—and what else is probability for, if not integration? -- you probably don’t want surreal numbers, because limits are not going to work there. (Some things that are normally done with limits can be recovered for surreals by other means, e.g. there’s a surreal exponential, but you don’t define it as a limit of partial sums, because that doesn’t work. So, maybe you can develop the necessary theory based on something other than limits, but I’m pretty sure it’s not something that already exists which you can just pick up and use.)
Again: Pick number systems for what they do. Hyperreals have a specific point, which is the transfer principle. If you’re not going to be using the transfer principle, you probably don’t want hyperreals. And as I already said, if you’re going to be taking any sort of limit, you probably don’t want surreals.
Consider asking whether you need a system of numbers at all. You mention sequences of real numbers; perhaps that’s simply what you want? Sequences of real numbers, not modulo a free ultrafilter? You don’t need to use an existing system of numbers, you can purpose-build one; and you don’t need to use a system of numbers at all, you can just use appropriate objects, whatever they may be. (Oftentime it makes more sense to represent “orders of infinity” by functions of different growth rates—or, I guess here, sequences of different growth rates.)
(Honestly if infinitesimal probabilities or utilities are coming up, I’d consider that a flag that something has likely gone wrong—we have good reasons to use real numbers for these, which I’m sure you’re already familiar with (but here’s a link for everyone else :P ) -- but I’ll admit that I haven’t read this thing in any detail and you are going beyond that sort of classical context so, hey, who knows.)
So, I haven’t really read this in any detail, but—I am very, very wary of the use of hyperreal and/or surreal numbers here. While as I said I haven’t taken a thorough look at this, to me these look like “well we need infinitesimals and this is what I’ve heard of” rather than having any real reason to pick one of these two. I seriously doubt that either is a good choice.
Hyperreals require picking a free ultrafilter; they’re not even uniquely defined. Surreal numbers (pretty much) completely break limits. (Hyperreals kind of break limits too, due to being of uncountable cofinality, but not nearly as extensively as surreal numbers do, which are of proper-class cofinality.) If you’re picking a number system, you need to consider what you’re actually going to do with it. If you’re going to do any sort of limits or integration with it—and what else is probability for, if not integration? -- you probably don’t want surreal numbers, because limits are not going to work there. (Some things that are normally done with limits can be recovered for surreals by other means, e.g. there’s a surreal exponential, but you don’t define it as a limit of partial sums, because that doesn’t work. So, maybe you can develop the necessary theory based on something other than limits, but I’m pretty sure it’s not something that already exists which you can just pick up and use.)
Again: Pick number systems for what they do. Hyperreals have a specific point, which is the transfer principle. If you’re not going to be using the transfer principle, you probably don’t want hyperreals. And as I already said, if you’re going to be taking any sort of limit, you probably don’t want surreals.
Consider asking whether you need a system of numbers at all. You mention sequences of real numbers; perhaps that’s simply what you want? Sequences of real numbers, not modulo a free ultrafilter? You don’t need to use an existing system of numbers, you can purpose-build one; and you don’t need to use a system of numbers at all, you can just use appropriate objects, whatever they may be. (Oftentime it makes more sense to represent “orders of infinity” by functions of different growth rates—or, I guess here, sequences of different growth rates.)
(Honestly if infinitesimal probabilities or utilities are coming up, I’d consider that a flag that something has likely gone wrong—we have good reasons to use real numbers for these, which I’m sure you’re already familiar with (but here’s a link for everyone else :P ) -- but I’ll admit that I haven’t read this thing in any detail and you are going beyond that sort of classical context so, hey, who knows.)