The thing that is special about R is that it is the unique (up to relabelling) complete ordered Archimedean field.
You’ve already said that it has to be ordered. It also needs to be a field so we can add, multiply, and divide things corresponding to the various properties that arise from Cox’s theorem. Completeness is exactly the “no holes” property.
The final one is the Archimedean property. This one says that for any element x>0 there is some natural number n such that nx>1. In terms of plausibility, it says that anything “infinitesimally small” in plausibility should be represented by zero, and not one of some infinite set of things that are for all finite purposes equivalent to zero.
If you relax the Archimedean property, you get systems such as hyperreals, surreals, and such. They all necessarily extend R, and so you do still end up needing to deal with R anyway. They also have awkward properties when used as numbers by which to measure things (such as countable additivity failing to be sufficient, but uncountable additivity being too restrictive to work).
The thing that is special about R is that it is the unique (up to relabelling) complete ordered Archimedean field.
You’ve already said that it has to be ordered. It also needs to be a field so we can add, multiply, and divide things corresponding to the various properties that arise from Cox’s theorem. Completeness is exactly the “no holes” property.
The final one is the Archimedean property. This one says that for any element x>0 there is some natural number n such that nx>1. In terms of plausibility, it says that anything “infinitesimally small” in plausibility should be represented by zero, and not one of some infinite set of things that are for all finite purposes equivalent to zero.
If you relax the Archimedean property, you get systems such as hyperreals, surreals, and such. They all necessarily extend R, and so you do still end up needing to deal with R anyway. They also have awkward properties when used as numbers by which to measure things (such as countable additivity failing to be sufficient, but uncountable additivity being too restrictive to work).