The relevant distinction isn't between "bounded" and "finite", but between the size of a container and the size of its elements. The difference between "bounde

The relevant distinction isn't between "bounded" and "finite", but between the size of a container and the size of its elements. The difference between "bounded" and "infinite" is the difference between "measure" and "cardinality". That is , "bounded" refers to a notion of distance, while "finite" refers to a measure of count. There are bounded infinities (such as the set of all rational or numbers between 0 and 1), but there are no unbounded finities. A container that contains finite objects of all sizes must be an infinite/unbounded container.