Let $S$ be a set. A total order on $S$ is a binary relation $\le$ on $S$, equivalently a subset of $S\times S$, and we write $x\le y$ to mean that $(x,y)\in\le$. The relation $\le$ is a total order if the following axioms hold.

1. For every $x\in S$, one has $x\le x$. [Reflexivity]
2. For all $x,y\in S$, if $x\le y$ and $y\le x$, then $x=y$. [Antisymmetry]
3. For all $x,y,z\in S$, if $x\le y$ and $y\le z$, then $x\le z$. [Transitivity]
4. For all $x,y\in S$, one has $x\le y$ or $y\le x$. [Comparability or totality]