A partial order is complete (or an -cpo) just if it has least upper bounds for all -chains.
An -chain is just an increasing sequence
of elements of . The 's can be thought of as a sequence of increasingly informative elements of , and the least upper bound can be thought of as the "limit" of this sequence, i.e. the least informative object that contains all the information carried by the approximations .
The useful thing about cpos if that every -continuous map has a least fixed point. This is known as Kleene's fixed point theorem.