Set Theory

Let $(X,\leq)$ be an ordered set. $X$ is linearly ordered if all points are comparable, i.e. for all $x,y\in X$ we have either $y\leq x$ or $x\leq y$. $X$ is well-ordered if every subset $A$ of $X$ has a minimal element. $X$ is said to be a completely ordered if all linearly ordered subsets of $X$ have a supremum in $X$. We say $X$ is inductively ordered if every linearly ordered subset of $X$ has an upper bound.