Motion P1788/M003.01_Set_of_reals NO
NO
I have two problems with the definitions. An interval should be an abstraction. A set of reals is one possible incarnation, but so is a set a probability distributions all of whose supports are constrained by the endpoints. It doesn't seem like good mathematics to be unnecessarily specific.
I agree with most of Ian Macintosh's views, except to say that, from the perspective of non-standard analysis, it seems that we should think of Infinity as a set of values itself rather than a single value, that is, something representing many possible values from which no particular value can be determinied. This goes for infinitesimals too, such as 1/Infinity, which are useful in defining open and partially open intervals.
Scott Ferson