First, define an equivalence relation, $\sim$, such that two sets, $A$ and $B$ are equivalent, $A\sim B$ if and only if there exists a bijection between them. 
Then define
$$0=[\emptyset]\_\sim$$
Where I have used $[]\_\sim$ to denote the equivalence class. Then, to define the successor function,
$$S([n]\_\sim)=[n\cup \{n\}]\_\sim $$
Then,
$$\exists!\mathbb{N}(0\in\mathbb{N} \wedge(n\in \mathbb{N}\rightarrow S(n)\in \mathbb{N}))$$
And of course the set $\mathbb{N}$ is the natural numbers. Now that I've written this out, I feel like the axiom of infinity would have to be replaced with the last line. But my question is, is this a valid construction under the ZF system or does it require different axioms? I also fear that I may have violated Russell's paradox...