Exercise A0.V.3.04

Proposition

Every countable set can be well-ordered.

Proof

Standard order relation on $\mathbb{N}$ is a well-order. A countable set $S$ is isomorphic to $\mathbb{N}$ and if $f : S \to \mathbb{N}$ is a bijection, then we can define a well-ordering on $S$ by $a \leq b \iff f(a) \leq f(b).$

In the concrete case of $S = \mathbb{Z},$ we can define $f(0) = 0,$ $f(-n) = 2n - 1$ and $f(n) = 2n$ for $n > 0.$ This is clearly a bijection.

$\blacksquare$