A countable discrete measurable space is standard Borel

Define $d(s,t)=0$ when $s=t$ and $d(s,t)=1$ when $s\mathrm{\ne }t$. These values are nonnegative, the definition is symmetric, and $d(s,t)=0$ exactly when $s=t$. If $s\mathrm{\ne }u$, then at least one of $s\mathrm{\ne }t$ or $t\mathrm{\ne }u$ holds, so $d(s,u)=1\le d(s,t)+d(t,u)$; if $s=u$, the triangle inequality follows from nonnegativity. Thus [F3] makes $d$ a metric. For every $s\in S$, [F4] gives $B(s,1\mathrm{/}2)=\{s\}$, so [F5] shows that every subset of $S$ is open.

If $(s_n)$ is Cauchy, [F6] with the rational $ϵ=1\mathrm{/}2$ gives an index $N$ such that $d(s_m,s_n)<1\mathrm{/}2$ for all $m,n\ge N$. Hence all $s_n$ for $n\ge N$ are equal, so the sequence converges in $S$; by [F7], $d$ is complete. Since $S$ is at-most-countable and dense in itself, [F8] makes $(S,d)$ separable.

The power set is closed under relative complements and countable unions, so it is a sigma-algebra by [F2]. By [F10] and step 1.1, every subset of $S$ belongs to the Borel sigma-algebra. Conversely, every Borel set is a subset of $S$, so this sigma-algebra is exactly $\mathcal{P}(S)$. Therefore $(S,d)$ is Polish by [F9], and the identity map from $(S,\mathcal{P}(S))$ to $(S,\mathcal{B}(S,d))$ is a measurable bijection with measurable inverse.

The identity map in step 3.1 is a measurable isomorphism to a Polish space, so [F11] gives that $(S,\mathcal{P}(S))$ is standard Borel.