Non-traditional intervals and their use. Which ones really make sense?

The paper discusses the question of why intervals, which are the main object of Interval Analysis, have exactly the form that we know well and habitually use, and not some other. In particular, we investigate why traditional intervals are closed, i.e. contain their endpoints, and also what is wrong with an empty interval. A second question considered in the work is how expedient it is to expand the set of traditional intervals by some other objects. We show that improper (“reversed”) intervals and the arithmetic of such intervals (the Kaucher complete interval arithmetic) are very useful from many different points of view.