Does any set admit a total order?

Question

Is it true that any set $P$ can be endowed with a total order $"\leq" \subseteq P\times P$?

Answer 1

Yes, in fact any set admits a well ordering. This fact is equivalent to the Axiom of Choice.

Of course the empty set is excluded.

For finite sets, axiom of choice is not needed. For infinite sets, the idea is, roughly, pick $x_1\in S$ where $S$ is an inifinte set, then pick $x_2\in S\setminus\{x_1\}$, and $x_3\in S\setminus\{x_1,x_2\}$ and so on.