What is the difference between an Ordered Set and a Completely Ordered Set?

Question

When is a set called an Ordered Set and when is it called a Completely Ordered Set?

Answer 1

A completely ordered is another phrasing for a totally ordered or linearly ordered set, i.e. an ordered set for which any two elements can be compared.

As an example, $\mathbf R$ is a totally ordered set (the order can be defined by the relation $x\le y\iff y-x\ge 0$).

Given a set $E$, we may consider the set $\mathcal P(E)$ of its subsets, and order it by inclusion: we say $X\prec Y$ if every element of $X$ is an element of $Y$, i.e. if $X\subset Y$. It is not a total order if $E$ has at least two elements, since if $a \ne b$, neither $\{a\}$ not $\{b\}$ is contained in the other.

Another example is the set $\mathbf N$ of natural numbers: we say $n\prec p$ if $n$ is a divisor of $p$, i.e. if there exists a natural number $p'$ such that $p=np'$. Thus for instance we can't write $ 2\prec 3$ nor $3\prec 2$.