Hyperreals can be multiplied by, divided by, added to, or subtracted from ordinary real numbers (such as probabilities) in a natural manner.
Hyperreals are defined in such a way that all statements in first-order predicate logic that use only predicates from basic arithmetic and that are true if we quantify only over reals are also true if we extend the domain of quantification to include hyperreals.
The hyperreals can be constructed as countably infinite sequences of reals in such a way as to satisfy the above axioms.
For example, the following two hyperreals are, respectively, strictly larger and strictly smaller than the above couple:
We want to say that one hyperreal is larger than another if it is larger in at least "almost all" places, and we want the resulting ordering to be complete, so that for every two hyperreals a, b, it is the case that a > b, or b > a, or a = b.
that the arithmetical structure of the real numbers uniquely matches the geometrical structure of lines in space; and that other number systems, like Robinson's hyperreals, accordingly fail to fit the structure of space.
Like the more familiar Figure 1, this picture corresponds to a number system, this one known as the hyperreals H, discovered by Abraham Robinson around 1960.
thrown down asa challenge to Robinson's hyperreals.
And now we seem to have come full circle; for our original question was how we know that the reals, and not the hyperreals, are best suited to that organizing role.
If so, so be it --my aim here has not been to create ah air of inevitability around the hyperreals, but to dispel the one around the reals.