Completeness of the Natural Numbers Do natural numbers have completeness property ? So, calculus can be done on the set of natural numbers ? So, why can't natural numbers be represented as a continuous line? Also, books generally tend to write calculus can be done on real numbers, does this only mean that on the set of real numbers, one can always find unique limits ?
 A: There are many Abstract definitions of Completeness. The Least Upper Bound Property and Cauchy condition are two of them with reference to the real numbers. But all of this essentially means that the real line is void of "gaps". There are no holes in it. Every point on the real line is a real number. This is what we derive from the completeness properties of $\Bbb R$. 
And yes I guess you are right. The set of natural numbers satisfies the supremum property and hence can be claimed to be complete. But the set of natural numbers is not dense. It is actually discrete. There are neighbourhoods of every natural number such that they contain no others. But the fact that the real line is without holes and gaps (or according to Dedekind's definition of continuity " If there is a point that splits the real numbers there is one and only one such point ") is a consequence of the completeness property of $\Bbb R$. But the so-claimed "completeness" of $\Bbb N$ obviously does not render any such implication and this is why the concepts of calculus and infinitesmal analysis work on the set of real numbers and not on the set of naturals. 
