I agree with all of that, but I think we should work out what decision theory actually needs and then use that. Surreals will definitely work, but if hyperreals also worked then that would be a really interesting fact worth knowing, because the hyperreals are so much smaller. (Ditto for any totally ordered affine set).
On second thoughts, I think the surreal numbers are what you want to use for utilities. If you choose any subset of the surreals then you can construct a hypothetical agent who assigns those numbers as utilities to some set of choices. So you sometimes need the surreal numbers to express a utility function. And on the other hand the surreal numbers are the universally embedding total order, so they also suffice to express any utility function.
I agree with all of that, but I think we should work out what decision theory actually needs and then use that. Surreals will definitely work, but if hyperreals also worked then that would be a really interesting fact worth knowing, because the hyperreals are so much smaller. (Ditto for any totally ordered affine set).
On second thoughts, I think the surreal numbers are what you want to use for utilities. If you choose any subset of the surreals then you can construct a hypothetical agent who assigns those numbers as utilities to some set of choices. So you sometimes need the surreal numbers to express a utility function. And on the other hand the surreal numbers are the universally embedding total order, so they also suffice to express any utility function.