[btilly]

In constructivism, once you have a Cauchy sequence of rationals, that represents a real. However given two such Cauchy sequences, you cannot always prove whether one is less than, equal to, or greater than the other.

This has some interesting consequences. For example only continuous functions can be functions in constructivism. (If you try to construct a function that is discontinuous at a point, there are Cauchy sequences you can give it that you cannot assign to a Cauchy sequence coming out. So it is not a well-defined function.)